Quantitative null-cobordism

By Gregory R. Chambers, Dominic Dotterrer, Fedor Manin, and Shmuel Weinberger, with an appendix by Fedor Manin and Shmuel Weinberger

Abstract

For a given null-cobordant Riemannian -manifold, how does the minimal geometric complexity of a null-cobordism depend on the geometric complexity of the manifold? Gromov has conjectured that this dependence should be linear. We show that it is at most a polynomial whose degree depends on . In the appendix the bound is improved to one that is for every .

This construction relies on another of independent interest. Take and to be sufficiently nice compact metric spaces, such as Riemannian manifolds or simplicial complexes. Suppose is simply connected and rationally homotopy equivalent to a product of Eilenberg–MacLane spaces, for example, any simply connected Lie group. Then two homotopic -Lipschitz maps are homotopic via a -Lipschitz homotopy. We present a counterexample to show that this is not true for larger classes of spaces .

1. Introduction

This paper is about two intimately related problems. One of them is quantitative algebraic topology: using powerful algebraic methods, we frequently know a lot about the homotopy classes of maps from one space to another, but these methods are extremely indirect, and it is hard to understand much about what these maps look like or how the homotopies come to be. The other is the analogous problem in geometric topology. The paradigm of this subject since immersion theory, cobordism, surgery, etc., has been to take geometric problems and relate them to problems in homotopy theory and, sometimes, algebraic K-theory and L-theory, and to solve those algebraic problems by whatever tools are available. As a result, we can solve many geometric problems without understanding at all what the solutions look like.

A beautiful example of this paradoxical state of affairs is the result of Nabutovsky that, despite the result of Smale (proved inter alia in the proof of the high-dimensional Poincaré conjecture) that every smooth codimension sphere in the unit -disk () can be isotoped to the boundary, the minimum complexity of the embeddings required in the course of such an isotopy (measured by how soon normal exponentials to the embedding intersect) cannot be bounded by any recursive function of the original complexity of the embedding. Effectively, an easy isotopy would give such a sphere a certificate of its own simple connectivity, which is known to be impossible.

In other situations, such as those governed by an -principle, a hard logical aspect of this sort does not arise. In this paper we introduce some tools of quantitative algebraic topology which we hope can be applied to showing that various geometric problems have solutions of low complexity.

As a first and, we hope, typical example, we study the problem, emphasized by Gromov, of trying to understand the work of Thom⁠Footnote1 on cobordism. Given a closed smooth (perhaps oriented) manifold, the cobordism question is whether it bounds a compact (oriented) manifold. The answer to this is quite checkable: it is determined by whether the cycle represented by the manifold in the relevant (i.e., or ) homology of a Grassmannian (where the manifold is mapped in via the Gauss map classifying the manifold’s stable normal bundle) is trivial.

1

Thom solved the unoriented version of this exactly, and he only solved the rational version of the oriented question. However, later work of Milnor and Wall did the more difficult homotopy theory necessary for the oriented case.

This raises two questions: First, how is the geometry of a manifold reflected in the algebraic topological problem? Second, how difficult is it to find the null-homotopy predicted by the algebraic topology? As a test of this combined problem, Gromov suggested the following question: Given a manifold, assume away small scale problems by giving it a Riemannian metric whose injectivity radius is at least and whose sectional curvature is everywhere between and . These properties can be achieved through a rescaling. A manifold possessing these properties will be said to have bounded local geometry. The geometric complexity of such a manifold can be measured by its volume.

If is a smooth compact manifold, without a specified metric, we measure its (differential-) topological complexity by the infimum of the geometric complexity over all metrics with bounded local geometry. (If is not closed, we require it to look like a collar within distance 1 of the boundary.) This is a reasonable complexity measure: there are only finitely many diffeomorphism classes of manifolds with a given bound on complexity; see Reference Che70, Reference Pet84, Reference Gro98, §8D.

The central question is as follows. Given a smooth (oriented) manifold of complexity which is null-cobordant, what is the least complexity of a null-cobordism? That is, if is an (oriented) compact Riemannian -manifold of bounded local geometry, which bounds a manifold diffeomorphic to , how small can the volume of be? Gromov has observed Reference Gro96, §5 II that tracing through the relevant mathematics would give a tower of exponentials of (of size around the dimension of the manifold minus ), but he has suggested Reference Gro99 that the truth might be linear.

The linearity problem, if it has an affirmative solution, would require very new geometric ideas and seemingly a solution to the cobordism problem essentially different from Thom’s. We build on Thom’s work to obtain the following:

Theorem A.

If is an (oriented) closed smooth null-cobordant manifold of complexity , then it has a null-cobordism of complexity at most

The degree of this polynomial, obtained by tracing through our arguments, grows exponentially with dimension. In the Appendix, we improve this result to give an only slightly superlinear bound on the size of the null-cobordism. F. Costantino and D. Thurston have already shown that for 3-manifolds, one does not need worse than quadratic growth for the complexity of the null-cobordism⁠Footnote2 Reference CT08.

2

Though they use a PL measure of complexity, the number of simplices in a triangulation.

Our proof follows the ideas of Thom quite closely and is based on making those steps quantitative (if suboptimally) and then getting an a priori estimate on the size of the most efficient null-homotopy of a Thom map when the homological condition holds.

Thom’s work starts by embedding into a sphere (or equivalently Euclidean space). This is already an act of violence: one knows that this will automatically introduce distortion. This is one source of growth that we do not know how to avoid.⁠Footnote3

3

A proof of the nonoriented cobordism theorem was given by Reference BH81 without using embedding. However, at a key moment there is a “squaring trick” in the proof, which also ends up giving, as a result of an induction, a polynomial estimate with an -degree polynomial.

For manifolds embedded in the sphere, the Lipschitz constant of the Thom map is closely related to the complexity of the submanifold⁠Footnote4 and the thickness of a tubular neighborhood. Conversely, if we know something about the Lipschitz constant of a null-homotopy of the Thom map, we can extract a geometrically bounded transverse inverse image.

4

Thom produces the null-cobordism from a null-homotopy by taking a transverse inverse image.

Zooming in, we see three issues that need to be taken care of.

(1)

We need to bound the Lipschitz constants of the maps at time in a null-homotopy (its “thickness”). Gromov has suggested Reference Gro99 that these frequently have a linear bound for maps of finite complexes into finite simply connected complexes.⁠Footnote5

5

If the domain is a circle and the target is a 2-complex, then for manifolds with an unsolvable word problem, there can be no computable upper bound for the worst Lipschitz constant in a null-homotopy. But for many groups with small Dehn function, it is possible to do this with only a linear increase. In particular, simple connectivity is an extremely natural requirement.

(2)

Bounding the worst Lipschitz constant arising in a null-homotopy does not quite suffice. One needs to bound the width⁠Footnote6 of the null-homotopy as well. This is a nontrivial issue: a null-homotopy of thickness can in general be replaced by one of width where is the dimension of the domain, but this is the best “automatic” bound.

6

The Lipschitz constant in the time direction.

(3)

Even provided such bounds, a transverse inverse image may be very large compared to the original manifold.

We deal with (1) and (2) simultaneously; this is the homotopy-theoretic result mentioned earlier. The real loss in our theorem comes from (3). In order to find a quantitative embedding of our manifold into , we are forced to take to be very large, and the embedded submanifold has small support in the resulting sphere. However, the support of a null-homotopy may still be quite large. This problem of the increase in the support is also one we have made no progress on and which seems important in a context broader than just cobordism theory.

1.1. Building Lipschitz homotopies

The main technical result of the paper is the following:

Theorem B.

Let be an -dimensional finite complex, and let be a finite complex which is rationally equivalent to a product of simply connected Eilenberg–MacLane spaces through dimension . If are -Lipschitz homotopic maps, then there is a homotopy between them which is -Lipschitz as a map from to .

The simplest settings in which this theorem applies are those in which is an odd-dimensional sphere or in which is a -sphere and . More generally, may be any Lie group or, even more generally, H-space. Given that the targets in many topological problems are H-spaces, we are optimistic that this partial result regarding the linearity of homotopies will have more general application. (We give an example below showing that this theorem cannot be extended to arbitrary simply connected complexes in place of .)

One antecedent to this result is given in Reference FW13, where maps with target possessing finite homotopy groups are studied. In that setting, the width of a null-homotopy is actually bounded universally, independent of . On the other hand, that paper shows that for any space with infinite homotopy groups there cannot be too uniform of an estimate of a linear upper bound on null-homotopies.

The obstruction in Reference FW13 has to do ultimately with homological filling functions. Isoperimetry, likewise, comes up in our result and is best appreciated by considering the following very concrete setting:

Lemma.

If is a degree map with Lipschitz constant , then there is a -Lipschitz null-homotopy for some .

This can be proved following the classical idea of Brouwer of cancelling point inverses with opposite local degree, but in a careful layered way so as to be able to control the Lipschitz constants. We will give a careful explanation of this as it provides the main intuition for the proof of Theorem B0.

1.2. Obstruction theory

Let be a null-homotopic -Lipschitz map. We assume this has a very particular structure; later we will see that such a structure can be obtained with only small penalties on constants. The domain sphere is a subdivision of a tetrahedron into grid isometric subsimplices, to a side. The map maps its 1-skeleton to the basepoint; for every 2-simplex either it also maps it to the basepoint or it maps a ball in the simplex homeomorphically to minus the basepoint, with degree .

To construct a null-homotopy of , we need to connect the positive and negative preimages with tubes in . Care must be taken to route these tubes in such a way that there are not too many clustered in any given spot, as in Figure 1. To do this, we decide beforehand how many tubes need to go through any particular part of and then connect them up in any available way.

To make this precise, assume that the tubes miss . Then we can count the number of tubes going through for each 1-simplex of . Every tube that goes into , for any 2-simplex , must either come out through another edge or come back to 0. In other words, if is the cochain which indicates the number of tubes (with sign!) going through , then gives the degree of on 2-simplices of . In the language of obstruction theory, is the obstruction to null-homotoping , and the existence of demonstrates that the obstruction can be resolved.

To ensure that it can be resolved efficiently, we need to pick a relatively small . The best we can do is to choose an which takes values . By considering a situation with degree on one side of canceling out degree on the other side, we see that we can do no better. That this is also the worst possible situation follows from the classical isoperimetric inequality for spheres; this is discussed in much greater generality in section 3.

In effect, once we have set , deciding how many tubes must go through a given point, we can connect them up in an entirely local way. We give a cellulation by prisms of length and base the 2-simplices of . We then construct the map by skeleta on this cellulation:

(1)

First, map the 1-skeleton to the basepoint.

(2)

Next, we can map the 2-cells via maps of degree between and in such a way that the map on the boundary of each prism has total degree , as in Figure 2(a). (It is here that we “layer” the null-homotopy.)

(3)

Finally, we choose a way to connect pairs of preimages on each prism via tubes, as in Figure 2(b). Since the number of tubes in each prism is bounded, we can do this with bounded Lipschitz constant.

For the second step, we need to use our . If we ensure that for each 1-simplex of , the degree of on is , then will have degree 0 on the boundary of each “long prism” , where is a 2-simplex of .

It remains to make sure that the degree is on the “short prisms”. To do this, we spread as evenly as possible along the unit interval: for every integer , the degree of on is . This then also determines the required degree on for every 2-simplex and time to make the total degree on the boundary of each prism . It is easy to check that the resulting degrees on all 2-cells are at most 3.

1.3. Outline of proof of Theorem B

We now describe how the proof of the above Lemma leads to the proof of Theorem B. The motto is the same: if we can kill the obstruction to finding a homotopy, then we can do the killing in a bounded way.

The first step is to reduce to a case where obstruction theory applies. For this, we simplicially approximate our map in a quantitative way. That is, given a map between metric simplicial complexes, the fineness of the subdivision of must be inversely proportional to the Lipschitz constant of the map.

From here the general strategy is to build a homotopy by induction on the skeleta of with a product cell structure. This homotopy will not in general be simplicial, but it will have the property that restrictions to each cell form a fixed finite set depending only on and . Every time we run into a null-cohomologous obstruction cocycle, we use a cochain that it bounds to modify the map on the previous skeleton. We ensure that these modifications are chosen from a fixed finite set of maps, leaving us with a fixed finite set of maps on the boundaries of cells one dimension higher. Then we can fill each such map in a fixed way, preserving the desired property.

When the obstructions are torsion, the main issue is the well-known one that killing obstructions “blindly” will sometimes lead to a dead end even when a homotopy exists. On the other hand, since there is a finite number of choices of torsion values for a cochain to take, we may avoid this by following a “road map” given by a known, but potentially uncontrolled, null-homotopy of . This is the content of Lemma 4.1.

On one hand, when we get integral obstructions, our choice of rational homotopy structure ensures that such issues do not come up. On the other hand, we do need to worry about isoperimetry. This is covered by Theorem 4.2, which generalizes the argument above.

2. Preliminaries

In this section, we discuss how to subdivide a metric simplicial complex so that the edges all have length approximately for a specified . We also show that, for any simplicial map and any , we can subdivide as above to form and homotope through a short homotopy to .

2.1. Regular subdivision of simplices

Definition.

Define a simplicial subdivision scheme to be a family, for every and , of metric simplicial complexes isometric to the standard with length 1 edges, such that restricts to on all faces. A subdivision scheme is regular if for each there is a constant such that has at most isometry classes of simplices and a constant such that all 1-simplices of have length in .

Given a regular subdivision scheme, we can define the -regular subdivision of any metric simplicial complex, where each simplex is replaced by an appropriately scaled copy of .

Note that times barycentric subdivision is not regular. On the other hand, there are at least two known examples of regular subdivision. One is the edgewise subdivision of Edelsbrunner and Grayson Reference EG00, which has the advantages that the -regular subdivision of is and that the lengths of edges vary by a factor of only . Roughly, the method is to cut the simplex into small polyhedra by planes parallel to the -dimensional faces, then partition each such polyhedron into simplices in a standard way. The other is described by Ferry and Weinberger Reference FW13: the trick is to subdivide into identical cubes, then subdivide these in the obvious way into cubes, and finally subdivide these in a canonical way into simplices. This method has the advantage of being easy to describe.

None of the listed advantages is crucial for our continued discussion, so we may remain agnostic as to how precisely we subdivide our simplices.

2.2. Simplicial approximation

Proposition 2.1 (Quantitative simplicial approximation theorem).

For finite simplicial complexes and with piecewise linear metrics, there are constants and such that any -Lipschitz map has a -Lipschitz simplicial approximation via a -Lipschitz homotopy.

Proof.

We trace constants through the usual proof of the simplicial approximation theorem, as given in Reference Hat01.

Denote the open star of a vertex by . Let be a Lebesgue number for the open cover is a vertex of of , that is, a number such that every -ball in is contained in one of the sets in the cover. Then is a Lebesgue number for the open cover of . Take a regular subdivision of so that for some each simplex of has diameter between and . Hence maps the closed star of each vertex of to the open star of some vertex of . This gives us a map which takes adjacent vertices of to adjacent vertices of , and hence if is the maximum edge length of , is -Lipschitz.

By a standard argument, this map extends linearly to a map with the same Lipschitz constant. The linear homotopy from to has Lipschitz constant .

Remark.

Suppose that and are -dimensional and made up of standard simplices of edge length 1. Then is the inradius of a standard simplex, and by using the edgewise subdivision, we can make sure that . Thus the Lipschitz constant of the map increases by a factor of at most

Furthermore, if is two-dimensional, then all of the edge lengths of the subdivision are equal. Therefore, in this case, and, in fact, it approaches for large , since we can choose a subdivision parameter very close to and, thus, very close to .

We will use simplicial approximation mainly as a way of ensuring that our maps have a uniformly finite number of possible restrictions to simplices. Almost all instances of “simplicial” in this paper can be replaced with “such that the restrictions to simplices are chosen from a finite set associated with the target space”. This formulation makes sense even when the target space is not a simplicial complex. In particular, it is preserved by postcomposition with any map, for example one collapsing certain simplices.

3. Isoperimetry for integral cochains

The goal of this section is to prove the following (co)isoperimetric inequality.

Lemma 3.1 ( coisoperimetry).

Let be a finite simplicial complex equipped with the standard metric, and let be the cubical or edgewise -regular subdivision of , and let . Then there is a constant such that for any simplicial coboundary there is an with such that .

We will start by proving the much easier version over a field; in the rest of the section will denote or . Then we will demonstrate how to find an integral-filling cochain near a rational or real one.

Lemma 3.2.

Let be a finite simplicial complex equipped with the standard metric, and let be an -regular subdivision of . Then for any , there is a constant such that for any simplicial coboundary , there is an with such that .

Proof.

We first show a similar isoperimetric inequality and then demonstrate that it is equivalent to the coisoperimetric version.

Lemma 3.3.

There is a such that boundaries of simplicial volume bound chains of simplicial volume at most .

Proof.

There are two ways we can measure the volume of a simplicial -chain in . The first, simplicial volume, is given by assigning every simplex volume 1, i.e.,

Alternatively, we can measure the -mass of chains: the mass of a simplex is its Riemannian -volume, and in general

Thus there are constants and , depending on the choice of subdivision scheme, such that for every -chain ,

Therefore to prove the lemma it suffices to show that a boundary whose -mass in is bounds a chain whose -mass is at most .

Our main tool here is the Federer–Fleming deformation theorem, a powerful result in geometric measure theory which allows very general chains to be deformed to simplicial ones in a controlled way. One proves this result by shining a light from the right spot inside each simplex so that the resulting shadow on the boundary of the simplex is not too large. By iterating this procedure on simplices of each dimension between and , we eventually end up with a shadow in the -skeleton, which is the desired simplicial chain. Federer and Fleming’s original version Reference FF60, Thm. 5.5 was based on deformation to the standard cubical lattice in . However, everything in their proof, except for the precise constants, translates to simplicial complexes. (See Reference EPC92, Thm. 10.3.3 for a proof of a slightly narrower analogue in the case of triangulated manifolds, which however also applies to any simplicial complex.)

Federer and Fleming’s theorem works for normal currents. To avoid this rather technical concept, we state the result for Lipschitz chains, that is, singular chains whose simplices are Lipschitz.

Theorem (Federer–Fleming deformation theorem).

Let be an -dimensional simplicial complex with the standard metric on each simplex. There is a constant such that the following holds. Let be a Lipschitz -chain in with coefficients in . Then we can write , where

can be expressed as an -linear combination of -simplices of

if can already be expressed as a combination of -simplices of (for example, if is a cycle), then and

Now suppose that is given a metric whose simplices are not standard, but such that the identity map satisfies

for all . When mass is measured with respect to , the bounds in the theorem become

(1)

;

(2)

;

(3)

.

We apply the theorem twice. First, we apply it to as a Lipschitz cycle in to show that it is homologous to a -cycle of volume via a Lipschitz -chain of volume . Next, we apply it to as a -chain in . Notice that the ratio is bounded independent of for a regular subdivision; therefore, deforms rel boundary to a chain in of volume , where depends on the subdivision scheme. Finally, bounds a chain in of volume , where depends only on the geometry of . Thus we can set .

Lemma 3.4.

Let be a finite simplicial complex. Then the following are equivalent for any constant :

any boundary has a filling with

any coboundary is the coboundary of some with .

The authors would like to thank Alexander Nabutovsky and Vitali Kapovitch for pointing out this simplified proof.

Proof.

The cochain complex is dual to the chain complex, and the -norm on cochains is dual to the volume norm on chains. So consider the general situation of a linear transformation between two normed vector spaces , and let be the operator norm of the transformation

where the norm of an equivalence class is given by . When is the boundary operator on , is exactly the minimal constant in condition (1). Hence this is also the operator norm of the dual transformation . It remains to investigate the dual norms on these spaces.

By the Hahn–Banach theorem, any operator on extends to an operator of the same norm on all of . Hence the dual norm of is exactly the norm on , and similarly the dual norm of is . Therefore, the operator norm of is the minimal constant of condition (2).

Combining Lemmas 3.3 and 3.4, we complete the proof of the rational and real versions of the coisoperimetry lemma.

Now we introduce the ingredients for proving the integral version.

Definition.

A -spanning tree of a simplicial complex is a -dimensional subcomplex which contains , such that the induced map

is an isomorphism and . A -wrapping tree of is a -dimensional subcomplex which contains and such that the induced maps

are both isomorphisms.

Lemma 3.5.

Every simplicial complex has a -spanning tree and a -wrapping tree.

Proof.

A -spanning tree for any can be constucted greedily starting from . At each step, we find a -simplex in such that represents a nonzero class in and add it to . Once there are no such simplices left, is an isomorphism. By construction, has no rational -cycles.

Notice that every -simplex of outside is a cycle in . To build a -wrapping tree from a -spanning tree, we may choose a basis for from among the simplices and add it to the tree.

Informally speaking, a -spanning tree should be thought of as the least subcomplex so that every -simplex outside is a cycle mod ; a -wrapping tree is the least subcomplex so that every -simplex outside is a boundary mod . In both cases the minimality means that there is a unique “completion” for a -simplex , i.e., a chain supported in (resp., ) so that is a cycle (resp., boundary).

Such spanning trees have been previously studied by Kalai Reference Kal83 and Duval, Klivans, and Martin Reference DKM09 and Reference DKM11 in the case where is the dimension of the complex. In that case the -simplices not contained in a spanning tree form a basis for (and a -wrapping tree is simply the whole complex). When contains simplices in dimension , however, there may be relations between the simplices when viewed as cycles in modulo . The next definition attempts to quantify the extent to which such relations constrain the behavior of cocycles in the pair .

Definition.

Let be a -spanning tree of . Consider the set of vectors in which are images of -simplices of . We define the gnarledness

We say that is -gnarled; we say a basis is optimal if is minimal in it.

The gnarledness measures the extent to which certain simplices are homologically “larger” than others. For example, consider a two-dimensional simplicial complex which is homeomorphic to the mapping telescope of a degree self-map of ,

Let us say we take a one-dimensional spanning tree which includes all but one of the simplices of both and ; let and , respectively, be the relevant 1-simplices in . Then in , and . For any basis for in which is a lattice point, , so the tree is at least 2-gnarled. Indeed, the same will happen for any spanning tree of this complex.

Lemma 3.6.

The cubical and edgewise -regular subdivisions of both admit -spanning trees which are at most -gnarled; the gnarledness is bounded independent of .

We will actually show this for grids in a cube complex. It is routine to modify this proof to work for the cubical subdivision of a simplicial complex; a similar construction works for the edgewise subdivision, since it consists of a grid of subspaces parallel to the faces which is then subdivided in a fixed way depending on dimension.

We first show that the subdivision of a cube has a -spanning tree rel boundary” with good geometric properties. To be precise:

Lemma 3.7.

Let be cubulated by a grid of side length , and let . We refer to

cells, i.e., faces of the cubulation;

faces, i.e., subcomplexes corresponding to faces of the unit cube; and

boxes, i.e., subcomplexes which are products of subintervals.

Then there is a -subcomplex of with the following properties:

.

deformation retracts to .

Every -cell of is homologous rel to a chain in whose intersection with each -face is a box.

More generally, every -dimensional box in is homologous rel to a chain in whose intersection with each -face is a box.

This subcomplex is illustrated in low dimensions in Figure 3.

Suppose now that we equip every face of in dimensions with subcomplexes satisfying these properties, and we let be the union of all these subcomplexes. Then by induction using property (4), any -cell of is homologous rel to a union of at most boxes in the -faces of . In turn, by property (2), each of these boxes has at most one cell outside . Therefore any -cell is homologous rel to a sum of at most cells in the -faces. This is the property that we use to prove Lemma 3.6.

Proof.

We construct by induction on and . For we can set . Similarly, for we can take to be less the interior of any one cell—for concreteness, let that be the cell that includes the origin.

Now we construct for by induction on . Write ; then for every , let , and for every , let

Finally, we throw in . It remains to show that the resulting complex satisfies the lemma.

It is clear that contains and that condition (1) holds. Moreover, deformation retracts first to using a retraction of , and thence to via a retraction of each layer individually. This demonstrates (2). It remains to show that (3) and (4) hold.

In order to do this more easily, we present an alternate rule for determining whether a -cell is contained in . Showing that it is indeed equivalent to the previous definition is tedious but straightforward. Let be a -cell of and let be the set of directions in which it has positive width. Write for the projection of onto its th interval factor, and write for the greatest integer such that . Then is in if and only if for some . In particular, if , then .

Now let be a -cell of . If , then it already fits the bill, so suppose it is in . We will argue that is bounded rel by a box with positive width in directions . Specifically, the projections of onto each interval factor of are as follows:

By the criterion above, contains only one cell which is not in , namely . Thus is the chain desired for (3).

More generally, given a -dimensional box in , one can take the union of the ’s constructed for each cell in the box. This gives a solution for (4).

Proof of Lemma 3.6 for cubulations.

Let be the complex obtained by dividing into grids at scale . We begin by choosing a -spanning tree for , then use it to build a -spanning tree for . We include all cells of contained in ; for every cell of not contained in , we include a complex as in Lemma 3.7. The resulting subcomplex includes and, by induction on , deformation retracts to . Therefore it is a -spanning tree for .

Now let be an optimal basis for . By the argument above, any -cell of is homologous rel to a sum of at most cells in the -faces, where is the dimension of . In turn, any -cell of which is contained in a -face represents the same homology class modulo as that face does modulo , and therefore can be represented as a sum of at most elements of . Therefore, .

We now have the tools we need to prove Lemma 3.1. We will do this by way of two auxiliary lemmas. The first states that any cochain with coefficients in which can be lifted to can be lifted to a cochain which is not too big.

Lemma 3.8 (Bounded lifting).

Let be a finite simplicity complex. Fix a cocycle and a -spanning tree of . Then if lifts to a cocycle , we can find such a lift with .

Proof.

Fix , a -wrapping tree of . Then for every -simplex of , there is a unique -chain supported in which fills mod . Moreover,

is a basis for : they are linearly independent since their boundaries are linearly independent in , and any -simplex in can be expressed as an integral linear combination . We can therefore extend by linearity to an isomorphism .

Now let be an optimal basis for which demonstrates that is -gnarled. For every , choose a representing it, and let

These are cycles and form a basis for .

Now for any cocycle and any -simplex of , we can write

The chain is a cycle, and hence homologous to the sum of at most elements of (with signs). Thus is determined by its values on . Conversely, any function extends to a -cocycle on : the values on determine its values on simplices of , while the values on determine its values on cycles. Since there are no cycles in , these are independent.

Now let be any lift of to a cocycle in . If we change by changing the values on by integers, we get a new cocycle; in particular, we can do this to get a new such that its values on are in . Now, for every -simplex , where the sum is over elements . Therefore, is still a lift of and has .

Now we show that if a chain has a filling with coefficients, we can find such a filling near any filling with coefficients.

Lemma 3.9.

Let be a finite simplicial complex equipped with the standard metric, let be the cubical or edgewise -regular subdivision of , and let . Then there is a constant such that for any , such that takes integer values and is a coboundary over , there is an such that and .

Proof.

By Lemma 3.6, admits a spanning tree whose gnarledness is bounded by a constant . Then by Lemma 3.8, the cocycle has a lift with . Then we can set and .

Proof of Lemma 3.1.

If , we can take , so suppose .

By Lemma 3.2, we can find an which satisfies and . Then by Lemma 3.9 we can find an such that and

This gives us an estimate for the isoperimetric constant .

4. Building linear homotopies

In this section we prove Theorem B. The proof is based on two lemmas: one to take care of obstructions posed by finite homotopy groups, and the other for infinite obstructions.

We start with a fairly general result for finite homotopy groups. It shows that if a map can be retracted to a subspace with finite relative homotopy groups, then one can force this retraction to be geometrically bounded. The special case in which is a point is proven in Reference FW13, Theorem 1.

Lemma 4.1.

Let be a pair of finite simplicial complexes such that is finite for . Then there is a constant with the following property. Let be an -dimensional simplicial complex, and let be a simplicial map which is homotopic to a map . Then there is a short homotopy of to a map which is homotopic to in , that is, a homotopy which is -Lipschitz under the standard metric on the product cell structure on .

Note that the constant does not depend on and in particular on the choice of a subdivision of . Thus if we consider Lipschitz and not just simplicial maps from to , the width of the homotopy remains constant, rather than linear, in the Lipschitz constant as is the case with some of our later results.

We will actually use the following relative version: if homotopes into rel , then there is a corresponding short homotopy rel . The proof below works just as well for this variant; one merely has to check that at every stage remains invariant.

Proof.

Let be a homotopy with and ; we have no control over this homotopy, only over . Our strategy will be to push both and the homotopy into via a second-order homotopy. Let be the 2-simplex with edges , , and opposite vertices , , and . At the end of the construction, we will obtain a map such that , lands in , and is the short homotopy we are looking for (see also Figure 4(a)).

We will construct this map one skeleton of at a time. At each step we ensure that the restrictions for simplices of are chosen from a finite set of Lipschitz maps depending only on and . In this way we get a universal bound on the Lipschitz constant. We start by setting and

In general, for , let

and . Then suppose by induction we have a map

such that the restrictions for -simplices of are contained in a finite set . We would now like to extend this (over cells of the form , for every -simplex of ; see Figure 4(b)) to a map .

To avoid doing an extra ad hoc step we will use the convention . Let . Given a -simplex , let

We think of this as a map . It can be homotoped into rel boundary via a null-homotopy which is constant on the -coordinate and sends to , keeping the vertex constant. Therefore the map

homotopes rel boundary into . Moreover, the set of homotopy classes of maps homotoping into (more precisely, of maps

which restrict to on ) is in (noncanonical) bijection with . One such bijection is obtained by sending a map to the map

which restricts to on and to everywhere else.

Now, by our inductive assumption, the number of different possibilities for the map is bounded above by

Let contain one Lipschitz map

for each possible value of and each homotopy class of null-homotopy; thus there are at most

such maps. We then set to be the element of for which . With this choice, the map can be extended in some way to . Since this part of the map does not need to be controlled, we can do this in an arbitrary way.

At the end of the induction, we have our map : the Lipschitz constant of is at most .

Now we prove Theorem B in the case where the target space is an Eilenberg–MacLane space. This will also be incorporated into the proof of the general case.

Theorem 4.2.

Let be a finite -dimensional simplicial complex, and let be a finite simplicial complex which has an -connected map , for some . Then there are constants and such that any two homotopic -Lipschitz maps are -Lipschitz homotopic through -Lipschitz maps.

This theorem is the main geometric input into the proof of Theorem B and is by itself enough to prove certain important cases. For example, it shows directly that any -Lipschitz map is -null-homotopic, as is any null-homotopic -Lipschitz map for any -dimensional . The general proof strategy is that described in §1.2.

Proof.

is homotopy equivalent to the CW complex obtained from it by contracting an -spanning tree. In order to create maps that we can homotope combinatorially, we simplicially approximate and on an -regular subdivision of and then compose with this contraction. After the homotopy is constructed, we can compose with the homotopy equivalence going back to get to the original . This increases constants multiplicatively and adds short homotopies to the ends; both of these can be absorbed into .

For the rest of the proof we assume that is the contracted complex and that and are compositions of simplicial maps with the contraction.

We construct the homotopy by induction on skeleta of . In particular . Let be the isoperimetric constant from Lemma 3.1, and let be the polyhedral complex given by the product cell structure on , where is split into subintervals . We define

by letting , , and sending the rest to .

Now define a simplicial cocycle by setting

for -simplices of . Since has a finite number of cells, there is a finite number of possible values of on simplices. In particular, for some .

By assumption, since , is a coboundary. By Lemma 3.1, for some cochain with . We will use to construct a cochain which we will use to extend to .

Define an extension of by

Clearly, is a cocycle. Moreover, since , one can see that

In particular the bound depends only on .

For each possible value of on cells, choose representatives

and extend to each -cell of using the appropriate representative to get . By construction, for each -cell of , is null-homotopic.

Now suppose we have constructed for some . By induction, there is a finite number, depending only on and , of possible restrictions , where is a -cell of . Moreover, if , is null-homotopic since . Thus for each possible restriction , we can choose an extension to . Extending to in this way gives us a finite set, depending on and , of possible restrictions to -cells.

At the conclusion of the induction, we obtain a map which is the desired null-homotopy.

In general, the constant increases by a multiplicative factor in each dimension, depending on the topology of . It is worth attempting to analyze and in simple cases, for example for maps . Here, simplicial approximation multiplies the Lipschitz constant by slightly more than . The induction has one step, and if satisfies , then satisfies . With a bit of care in plumbing as we connect preimages of on the surface of our 3-cells, we can build the null-homotopy by increasing the Lipschitz constant by a factor of 3. This gives a total multiplicative factor of when is large. The isoperimetric constant depends on the exact geometric model for the preimage sphere; in the case of the tetrahedron, it is 1.

Putting together Lemma 4.1 and Theorem 4.2, we can now prove Theorem B. We recall this result below:

Theorem.

Let be an -dimensional finite complex. If is a finite simply connected complex which is rationally equivalent through dimension to a product of Eilenberg–MacLane spaces, then there are constants and such that homotopic -Lipschitz maps from to are -Lipschitz homotopic through -Lipschitz maps.

A corollary for highly connected follows from the rational Hurewicz theorem.

Corollary 4.3.

Let be a rationally -connected finite complex, and let be an -dimensional finite complex. Then if , then there are constants and such that homotopic -Lipschitz maps from to are -Lipschitz homotopic through -Lipschitz maps.

Before giving the proofs of the corollary and the theorem, we recall some facts about maps to Eilenberg–MacLane spaces which derive from properties of the obstruction-theoretic isomorphism

induced by cell-wise degrees on cellular maps. See for example Reference Spa81, Chapter 8 for details. Let be any CW complex, let , and let be an abelian group, and consider a CW model of whose -skeleton is a point . Then:

is an H-space: the element in sending induces a multiplication map . This has identity , i.e., it sends

and is associative and commutative up to homotopy. It can also be assumed cellular.

Let be a map. Then the group

of self-homotopies of is naturally isomorphic to .

Denote the map that sends to also by . Then

acts freely and transitively on via the multiplication map; the above isomorphism takes this to the action of on itself via multiplication.

Proof of Corollary 4.3.

The rational Hurewicz theorem (see, e.g., Reference KK04) states that if is a simply connected space such that for , then the Hurewicz map

induces an isomorphism for . Therefore, for ,

In particular, we can find a map which induces the identity on . Then the map

is rationally -connected. This allows us to apply Theorem B.

Proof of Theorem B.

Suppose that is rationally homotopy equivalent through dimension to . This gives us a map inducing an isomorphism on . For each , let be in the preimage of the copy of corresponding to ; this induces a map . Then

is again a rational homology isomorphism, and so by the rational Hurewicz theorem, is a pair with finite for .

Let be homotopic -Lipschitz maps, and let . Then by Theorem 4.2, for each , there is a such that and are -Lipschitz null-homotopic through -Lipschitz maps via homotopies . Then

is a -Lipschitz homotopy. Suppose first that we can homotope to an uncontrolled homotopy of and in . Then by the relative version of Lemma 4.1 applied to the pair , there is a such that and are -Lipschitz homotopic in through -Lipschitz maps.

Note that such a homotopy may not exist a priori; we will need to modify so that it does. For this we use an algebraic construction. We know that there is some homotopy between and . So we can concatenate the homotopies and to give a map defined by

(where we think of as ) representing an element of . Since each factor is a high-dimensional skeleton of an H-space, there is a multiplication map for some large enough . This induces a free transitive action of on each .

We now analyze the cokernel of the group homomorphism

Consider the relative Postnikov tower

of the inclusion . Here, is a space such that for and for . The map therefore only has one nonzero relative homotopy group, . In this setting there is an obstruction theory long exact sequence ( Reference Bau77, §2.5; cf. also Reference GM81, Prop. 14.3 and Reference Sul74, Lemma 2.7) of groups

In particular, an element of is the image of some loop of maps to based at . Hence, independently of ,

always lifts to . Let be the (finite!) collection of linear combinations with coefficients between and of some finite generating set for . Then for any , the finite set

surjects onto the cokernel we are interested in.

We can then choose so that can be homotoped into . Now define a map by

Then is in the same homotopy class as . This means that the map given by is a homotopy between and which homotopes into and whose Lipschitz constant is bounded by

This is linear in , and except for , the coefficients depend only on , , and , so can be plugged into the argument above.

Remark.

Note that in this proof, the dependence of on lies only in the choice of generating set for . In certain special cases, this constant can be independent of . For example, suppose that we know that is an -sphere (or even just an -dimensional PL homology sphere). Then is generated by maps whose degree on simplices is at most 1—regardless of the geometry of . This means that for such homology spheres , -Lipschitz maps can be homotoped through maps of Lipschitz constant , though the width of the homotopy required may depend on the geometry. This may have applications such as finding skinny metric tubes between “comparable” metrics on the sphere. In contrast, results of Nabutovsky and Weinberger imply that without this comparability condition, such tubes may have to be extremely (uncomputably) thick.

5. A counterexample

One may ask whether the linear bound of Theorem B holds for any simply connected target space, not just products of Eilenberg–MacLane spaces. The answer is emphatically no. Here we give, for each , a space and a sequence of null-homotopic maps such that volume of any Lipschitz null-homotopy grows faster than the -st power of the Lipschitz constant of the maps. This forces the Lipschitz constant of the null-homotopy to grow superlinearly.

To make this precise: by the volume of a map , we mean

(recall that by Rademacher’s theorem the derivative of a Lipschitz map is defined almost everywhere). By this definition,

To construct the space , we take and attach -cells via attaching maps which form a basis for . Note that by rational homotopy theory, is a free graded Lie algebra on two generators of degree 1 whose Lie bracket is the Whitehead product (see Reference GM81, Exercise 44 or Reference FHT12, §24(f), Example 1). In particular, if and are the identity maps on the two copies of , the iterated Whitehead product

with repeated times, represents a nonzero element of . Moreover, the map

is an -Lipschitz representative of . Thus in we can define a null-homotopy of by first homotoping it inside to for some map of degree , and then null-homotoping each copy of via a standard null-homotopy.

Since is not null-homotopic in , this standard null-homotopy must have degree on at least one of the -cells, giving a closed -form on such that . Now, suppose is some other null-homotopy of . Then gluing and along the copies of gives a map . Note that if any map had nonzero degree on cells, then the map on the boundary would be homotopically nontrivial. This shows that must have total degree on cells, in other words, that . Thus the volume of a null-homotopy of grows at least as .

In the sequel to this paper Reference Cha18, we show that for , this estimate is sharp, in the sense that we can always produce a null-homotopy whose Lipschitz constant is quadratic in the time coordinate and linear in the others.

6. Quantitative cobordism theory

The goal of the rest of the article is to prove Theorem A, which we recall below.

Theorem.

If is an oriented closed smooth null-cobordant manifold which admits a metric of bounded local geometry and volume , then it has a null-cobordism which admits a metric of bounded local geometry and volume

Moreover, can be chosen to be .

As described in the introduction, we will prove this theorem by executing the following steps. We begin by choosing a metric on such that has bounded local geometry and such that the volume of is bounded by twice the complexity of . We then proceed as follows:

(1)

We embed into for an appropriately large (depending on ) so that the embedding has bounded curvature, bounded volume, and has a large tubular neighborhood. We will use this map to embed the manifold into the standard round sphere while maintaining bounds on its geometry.

(2)

We show that the Pontryagin–Thom map from this sphere to the Thom space of the universal bundle of oriented -planes in (relative to the embedded manifold and its tubular neighborhood) has Lipschitz constant bounded as a function of and the volume of .

(3)

We analyze the rational homotopy type of the Thom space and determine that, up to dimension , it is rationally equivalent to a product of Eilenberg–MacLane spaces. Since is null-cobordant, this map is null-homotopic and so, as a result, we can apply Theorem B to conclude that there is a null-homotopy which has Lipschitz constant bounded as a function of and the volume of . This translates to a map from the ball with boundary to the Thom space with the same bound on the Lipschitz constant.

(4)

The proof is completed by simplicially approximating this map from the ball, then using PL transversality theory to obtain an -dimensional manifold, embedded in this ball, which fills and satisfies the conclusions of the theorem.

Throughout this section, we use the following notation. We write to mean that there is a constant , depending only on , such that . Similarly, we write to imply that there are constants and , again depending only on , such that . We define the same expression with analogously. Throughout this section we will also use to denote the volume of . Lastly, we will write to denote the Grassmannian of oriented -dimensional planes in and to denote the Thom space of the universal bundle over this Grassmannian. is given the standard metric, which induces a metric on . Furthermore, we denote by the basepoint of the Thom space .

We begin by explicitly defining what “bounded local geometry” means in Theorem A.

Definition.

Suppose that is a closed Riemannian manifold of dimension . Following Reference CG85, we say that has bounded local geometry if it has the following properties:

(B1)

has injectivity radius at least .

(B2)

All elements of the curvature tensor are bounded below by and above by .

The manifold satisfies if in addition it satisfies the following condition:

(B3)

The th covariant derivatives of the curvature tensor are bounded by constants . (The are defined once and for all, but we will not specify them.)

Conditions (B1)–(B3) taken together agree with the standard definition used by Riemannian geometers, except that we require explicit quantitative bounds. A theorem of Cheeger and Gromov Reference CG85, Thm. 2.5 states that for any given a metric on with can be -perturbed to with which satisfies (B3). In particular, . By rescaling, we get a metric with and

Therefore, for the rest of the proof we can assume that (B3) holds, with a constant multiplicative penalty on the volume of our manifold.

Finally, if has boundary, we say, following Reference Sch01, that it satisfies if (B1) holds at distance at least from the boundary, (B2) holds everywhere, and in addition the neighborhood of of width 1 is isometric to a collar . In particular, this implies that .

6.1. Embedding into

To begin constructing the embedding described in the first step, we first choose a suitable atlas of . A similar set of properties defines uniformly regular Riemannian manifols, a notion due to H. Amann (see, for example, Reference DSS16, p. 4). However, we require our quantitative bounds on the geometry of the maps to be much more uniform, depending only on the dimension; we also require that the charts can be partitioned into a uniform number of subsets consisting of pairwise disjoint charts.

Lemma 6.1.

Suppose that is a compact orientable -dimensional manifold with . There exists a finite atlas with the following properties, expressed in terms of constants , , and depending only on , as well as a natural number for some universal constant .

Every map in is the exponential map from the Euclidean -ball of radius to which agrees with the orientation of . Since the injectivity radius of is at least and , this is well-defined. We write

Here, is a geodesic ball of of radius , and is the Euclidean ball of radius in .

can be written as the disjoint union of sets of charts such that any pair of charts from the same have disjoint image.

When we restrict all the maps in to , they still cover .

The pullback of the metric with respect to every is comparable to the Euclidean metric, that is,

for every , for every , and where is the pullback of at .

The first and second derivatives of all transition maps are bounded by .

Proof.

As mentioned above, this list of properties is closely related to one used in the definition of a uniformly regular Riemannian manifold. Every compact manifold is uniformly regular, and it is known that a (potentially noncompact) orientable manifold with for some is uniformly regular; this is shown in Reference Ama15. This guarantees an atlas with properties similar, though not identical, to the above. We use a similar set of arguments to those compiled by Amann.

To begin, we cover by balls of radius . Since is compact, we require only finitely many balls to cover . Furthermore, by the Vitali covering lemma, we can choose a finite subset of these balls such that also cover , and such that are disjoint. We also have that the balls cover , and that these balls have radius .

Fix a ball for some . We would like to count how many other balls in intersect . Call these balls . Then and all lie inside , and all are disjoint. Since has bounded local geometry, the volume of is bounded above in terms of , and the volumes of and are bounded below in terms of . This yields an exponential bound on in terms of . As a result, the balls can be partitioned into sets of pairwise disjoint balls. We define these sets as . This proof is analogous to a standard proof of the Besicovitch covering lemma in .

For every with , is defined as follows. For every ball , the exponential map goes from the Euclidean ball of radius to ; furthermore, it can be chosen so that it agrees with the orientation of . These are exactly the charts that comprise . The first three properties that we desire are now satisfied.

Property (4) is part of Reference HKW77, Lemma 1. Indeed, all th derivatives of the metric tensor are also bounded by a constant depending only on and ; see Reference Eic91, Reference Sch01. This allows us to also bound the derivatives of the pullback of the Euclidean metric along transition functions between the charts. Property (5) follows immediately from this.

We will also need the following simple observation.

Lemma 6.2.

There is a function from to such that:

is monotonically increasing with and .

for all .

for all .

For every , there is some such that

for all .

We will now embed into so that we have control over its geometry. In particular, we will prove the following proposition.

Proposition 6.3.

Suppose that is a compact orientable -dimensional Riemannian manifold with volume and bounded local geometry. Then there is some (in particular depends only on ) such that is diffeomorphic to a submanifold with the following properties:

lies in a ball of radius .

The smooth map sending to , the oriented tangent space of at , has Lipschitz constant .

has a normal tubular neighborhood of size .

Proof.

We will use the chart constructed in Lemma 6.1 to define an embedding of into , with depending only on . By property (2), can be written as a disjoint union of sets of charts with disjoint images. The number of elements in each is since these disjoint images have volume . Let . We define -dimensional spheres in by the following properties:

(1)

has radius ;

(2)

every passes through the origin;

(3)

the center of every lies on the ray from the origin in the direction .

The radii of the spheres are between and , and the difference between any two of the radii is . An example of such a sequence of spheres is shown in Figure 5. We will refer to the antipode of the origin on each sphere as its “north pole”.

Define and . Fix a point . Our embedding will map to , where each , as follows. For every with , if is not in the image of any chart of , then we set . If not, then is in the image of exactly one chart in , . In this case, we set to be the point on given by composing with a map which is defined as follows: take the origin to the north pole of , and then map the geodesic sphere of radius in homothetically to the geodesic sphere around the north pole in of radius . Here is defined as in Lemma 6.2 and is the intrinsic diameter of .

Define a map by

Since is smooth and all its derivatives go to 0 at , this is a smooth map whose derivative has rank on . If the original charts in are , then we can write

Since is an atlas, for any , some is nonzero. On the other hand, at most one of is nonzero. This shows that is an immersion. Moreover, if , then for some chart and are in the image of that chart, and in fact . This shows that is injective. Since every is contained in a ball of radius around the origin, every point in is mapped to a point in of norm .

We have a natural set of oriented charts for the embedded manifold given by for each . Since the first and second derivatives of all of the transition maps are bounded , since has bounded derivatives, and since the radii of the balls are all bounded below by and above by , the first and second derivatives of all charts are . Moreover, since every point of is contained in for some and , and for , the first derivative of each chart is .

Combined with the property that the pullback of the metric of using each chart is comparable to the Euclidean metric, this shows that the map from to with its intrinsic Riemannian metric is bi-Lipschitz with constant .

Let us now consider the map as defined in the statement of Proposition 6.3. Fix a point , choose one of the above charts which covers , and define to be the unique point with . Choose unit vectors in such that

is an orthonormal set of vectors that spans the tangent plane of at . For any unit vector , consider

Since all first and second derivatives of are bounded above by , and since the first derivatives of are bounded from below by , all of these values are bounded by . Since the original vectors are orthonormal, for sufficiently small the distance in between the tangent plane at and the tangent plane at is . Since is -bi-Lipschitz, this completes the proof that is -Lipschitz.

Lastly, we want to show that has a normal tubular neighborhood of width . Suppose that and are two points on , and suppose and are normal vectors at and respectively, such that . We would like to show that .

Let be the angle between and . Consider a minimal-length geodesic , parametrized by arclength, between and ; and lie in the orthogonal -planes to this geodesic at and , respectively. The above arguments imply that the tautological embedding has second derivatives . Therefore, the second derivative of is .

Proposition.

Let . Then .

Proof.

Let be the plane spanned by and , and let and be orthogonal projections to and . Then:

the average over of is ;

and .

The bounds on the second derivative then imply that for every ,

Therefore,

and therefore .

Now let be a chart in some such that . Suppose first that . Then the properties of any imply that ; in particular, and so .

On the other hand, suppose that is not in . Suppose first that it is in but not . Here again the properties of any imply that . The same is true if is not in the image of any . Finally, if is in for some , then the properties of the imply that . In all these cases it must be the case that

This completes the proof that has a large tubular neighborhood.

Finally, we prove a lemma which allows us to embed into a round sphere.

Lemma 6.4.

Suppose that is an embedded submanifold of satisfying the conclusions of Proposition 6.3. Then there is an embedding into the round unit sphere such that

has a tubular neighborhood of width . Additionally, can be extended to a -Lipschitz diffeomorphism from this tubular neighborhood to a neighborhood of width of .

The map given by has Lipschitz constant . Here, is the map from to from Proposition 6.3.

Proof.

is contained in a ball of radius , and without loss of generality we may assume that this ball is centered at the origin. If we restrict the stereographic projection to , we obtain an embedded manifold of which satisfies all of the above properties.

6.2. Proof of Theorem A

To complete the proof of Theorem A, we use the embedding of in produced by combining Proposition 6.3 with Lemma 6.4. We begin by describing the Pontryagin–Thom map and by computing its Lipschitz constant.

We map into , the Thom space of the universal bundle of oriented -dimensional planes in , via a map defined as follows. Let . If is outside of the tubular neighborhood of of width (here the constant depending on is the same as that in Lemma 6.4), then it is mapped to (the basepoint of ). If not, then applying to produces a point in the tubular neighborhood of of width (this constant depending on is the same as that in Proposition 6.3). Hence, where and is a point in the oriented normal plane of at , and has length . Both and are unique. We then take

Since the map from Lemma 6.4 is Lipschitz with Lipschitz constant , the map from to the oriented normal plane of at is also Lipschitz with Lipschitz constant . If we assume that is at most half the critical radius of the tubular neighborhood, then the projection has Lipschitz constant . Furthermore, the tubular neighborhood of is dilated by a factor of when it is mapped to and the map has Lipschitz constant on the tubular neighborhood of width of . Hence, the Lipschitz constant of is .

By Reference MS74, Theorem of Thom, p. 215, the map is null-homotopic, since (and so and with the orientation induced by the charts as in the proof of Proposition 6.3 and the stereographic projection from Lemma 6.4) is null-cobordant. is -connected by Reference MS74, Lemma 18.1. We can assume, perhaps by adding extra “empty” dimensions, that and so .

By Corollary 4.3, since is a metric CW complex, there is a null-homotopy of with Lipschitz constant . This constant depends only on , and so there is a null-homotopy of of Lipschitz constant . This extends to a map from a ball of radius in to with Lipschitz constant .

We now observe that we can consider both and as finite simplicial complexes in the following sense. Since the result follows from standard arguments, we omit the proof.

Lemma 6.5.

There is a finite simplicial complex and a scale such that if we give each simplex the metric of the standard simplex of side length , then there is a -bi-Lipschitz function from to . Moreover, the image of the section of under this map is a subcomplex (and a simplicial submanifold) of .

Similarly, there is a finite simplicial complex and a scale such that if every simplex is given the metric of the standard simplex of side length , then there is a -bi-Lipschitz function from to . We can also choose so that is a homeomorphism from to .

Both and depend only on .

We can now consider the map given by . Since the maps are -bi-Lipschitz, is still bi-Lipschitz. With a slight abuse of notation, we will refer to by , by , and by . By using Proposition 2.1, we can subdivide the simplices of to form such that can be homotoped to a simplicial map from to with Lipschitz constant . We also know that the side lengths of the simplices in are . We will define to be the simplicial submanifold formed by applying on the -bundle of .

Clearly, is a PL manifold which is homeomorphic to . This is because the map was assumed to be a homeomorphism from the boundary of the ball to the boundary of the simplicial approximation of the ball. We will begin by perturbing to , a PL manifold embedded in . We want to have the following properties:

(1)

is an -dimensional PL manifold.

(2)

is homeomorphic to .

(3)

For every open -simplex of , is transverse to .

(4)

depends only on .

We can find such a PL manifold by perturbing using PL transversality theory. There are several standard references for this; see for example Reference RS72, Theorem 5.3. This theorem does not yield this result directly but can be adapted to do so.

We will use the transverse inverse image of to construct our filling. We know that is homeomorphic to from property (2). Furthermore, the fact that the map is simplicial combined with properties (1) and (3) implies that is an -dimensional PL manifold with boundary, and its boundary is . Furthermore, since the sphere, the ball, the simplicial approximations to them, and the embedded manifold are all orientable, from the discussion in Reference MS74, p. 210 we see that we also have that this manifold is orientable, and agrees with the orientation of its boundary (which is homeomorphic to ).

We now estimate the volume of . Since only depends on , the number of simplices of is . Since is a simplicial map, the intersection of with a given simplex belongs to a finite set of subsets which depends only on ; since the simplices are at scale , the -dimensional volume of this intersection is . Therefore, the volume of is , where is .

To build our manifold, we smooth out and . We can do this so that the volumes do not increase very much and so that , after smoothing, is diffeomorphic to . As above, since and depend only on , since is a finite complex, and since the side lengths of the simplices in are , this smoothing can be done so that the result has (including on the boundary). After dilating the smoothed version of by a factor which is , we have a compact oriented manifold with whose boundary is (orientation preserving) diffeomorphic to . The dilation increases the volume of the resulting manifold by a factor of , and so the result still has volume bounded by .

In particular, after the dilation has been performed, we obtain a manifold with bounded local geometry with volume bounded by , and which bounds a manifold diffeomorphic to with locally bounded geometry. Thus the complexity of the null-cobordism of is . Since is within a factor of of the complexity of , this completes the proof of the theorem.

Appendix A. The Gromov–Guth–Whitney embedding theorem

1. Summary

By using a different method of embedding manifolds in Euclidean space, the bound of Theorem A can be improved to achieve one tantalizingly close to Gromov’s linearity conjecture:

Theorem A.

Every closed smooth null-cobordant manifold of complexity has a filling of complexity at most , where for every .

As with the original Theorem A, this holds for both unoriented and oriented cobordisms.

Recall that the polynomial bound on the complexity of a null-cobordism follows from a quantitative examination of the method of Thom:

(1)

One embeds the manifold in , with some control over the shape of a tubular neighborhood.

(2)

This induces a geometrically controlled map from to the Thom space of a Grassmannian; one constructs a controlled extension of this map to .

(3)

Finally, from a simplicial approximation of this null-homotopy, one can extract a submanifold of which fills and whose volume is bounded by the number of simplices in the approximation.

Part (2) is the result of the quantitative algebraic topology done to control Lipschitz constants of null-homotopies. Abstracting away the method of embedding, we extract the following:

Theorem.

Let be an oriented closed smooth null-cobordant manifold which embeds with thickness in a ball in of radius ; that is, there is an embedding whose exponential map on the unit ball normal bundle is also an embedding. Then has a filling of complexity at most . (For unoriented cobordism, is sufficient.)

This is optimal in the sense that the asymptotics of the estimates in steps (2) and (3) cannot be improved. Then to prove Theorem A, we simply need the following estimate, which may also be of independent interest.

Theorem B.

Let be a closed Riemannian -manifold of complexity . Then for every , has a smooth -thick embedding into a ball of radius

This then implies that for every , has a filling of complexity at most

proving Theorem A.

The embedding estimate is in turn derived from a similar estimate of Gromov and Guth Reference GG12 for piecewise linear embeddings of simplicial complexes. The combinatorial notion of thickness used in that paper does not immediately translate into a bound on the thickness of a smoothing. Rather, in order to prove our estimate, we first prove a version of Gromov and Guth’s theorem, largely using their methods, with a stronger notion of thickness which controls what happens near every simplex. We then translate this into the smooth world using the following result.

Theorem C (Corollary of Reference BDG17, Thm. 3).

Every Riemannian -manifold of bounded geometry and volume is -bi-Lipschitz to a simplicial complex with vertices with each vertex lying in at most simplices. In particular, every smooth -manifold of complexity has a triangulation with vertices and each vertex lying in at most simplices.

The PL picture

In dimensions all PL manifolds are smoothable. Therefore Theorems A and C together imply that for , every PL null-cobordant manifold with vertices and at most simplices meeting at a vertex admits a PL filling with vertices and at most simplices meeting at a vertex, where the function satisfies for every . For , this complements the result of Costantino and D. Thurston Reference CT08 which gives bounded geometry fillings of quadratic volume with no restrictions on the local geometry of .

On the other hand, in high dimensions the PL cobordism problem is still open, and poses interesting issues since, unlike in the smooth category, is not an explicit compact classifying space for PL structures. We hope to return to this in a future paper.

So, is it linear?

Gromov’s linearity conjecture appears even more interesting now that we know that it is so close to being true. On the other hand, at least in the oriented case, linearity cannot be achieved by Thom’s method. Suppose that one could always produce “optimally space-filling” embeddings , that is, 1-thick embeddings in a ball of radius . Even in this case, an oriented filling would have volume .

Moreover, recent results of Evra and Kaufman Reference EK16 on high-dimensional expanders imply that, at least for simplicial complexes, the Gromov–Guth embedding bound is near optimal and space-filling embeddings of this type cannot be found. While -manifolds are quite far from being -dimensional expanders, it is possible that a similar or weaker but still nontrivial lower bound can be found. This would show that Thom’s method is not sufficient for constructing linear-volume unoriented fillings, either.

On the other hand, at the moment we cannot reject the possibility that it is possible to find linear fillings for manifolds by some method radically different from Thom’s. In particular, it is completely unclear how to go about looking for a counterexample to Gromov’s conjecture, although we believe that ideas related to expanders may play an important role.

2. PL embeddings with thick links

In Reference GG12 Gromov and Guth describe “thick” embeddings of -dimensional simplicial complexes in unit -balls, for . They define the thickness of an embedding to be the maximum value such that disjoint simplices are mapped to sets at least distance from each other. Reference GG12, Thm. 2.1 gives a nearly sharp upper bound on the optimal thickness of such an embedding in terms of the volume and bounds on the geometry.

This condition is insufficient to produce smooth embeddings of bounded geometry, because as thickness decreases, adjacent 1-simplices of length may make sharper and sharper angles. In this section we show that Gromov and Guth’s construction can be improved to obtain embeddings that also have large angles. Recall that the link of a -simplex inside a simplicial complex is the simplicial complex obtained by taking the locus of points at any sufficiently small distance from any point of in all directions normal to . This complex contains an -simplex for every -simplex of incident to . If is linearly embedded in , there is an obvious induced embedding . We show the following:

Theorem 2.1.

Suppose that is a -dimensional simplicical complex with vertices and each vertex lying in at most simplices. Suppose that . Then there are and and a subdivision of which embeds linearly into the -dimensional Euclidean ball of radius

with Gromov–Guth thickness and such that for any -simplex of , the induced embedding is -thick.

Proof.

The proof proceeds with the same major steps as in Reference GG12. We first show that a random linear embedding which satisfies the condition that all links are thick, while not having the right thickness, is sparse in a weaker sense: most balls have few simplices crossing them. Gromov and Guth then show that the simplices can be bent locally, at a smaller scale, in order to thicken the embedding; this produces a linear embedding of a finer complex. We note that if the scale is small enough, this finer, bent embedding also has thick links.

We write for and to mean . Following Gromov–Guth, we actually embed in a -ball with thickness ; for simplicity, write .

We start by choosing, uniformly at random, an assignment of the vertices of to points of from those such that for some , the following hold:

(1)

Adjacent vertices are mapped to points at least distance apart.

(2)

The linear extension to an embedding of has -thick links.

We call the resulting linear embedding . We can choose so that this is possible since the thickness of the link of some vertex (and of incident higher-dimensional simplices) only depends on the placement of vertices at most distance 2 away. Moreover, this implies the following:

()

The probability distribution of conditional on some prior distribution on the other vertices is pointwise the uniform distribution. This follows from the fact that this is true even when all vertices within distance 2 from are fixed.

This implies that given a -simplex , the probability distribution of (conditional on any distribution on the vertices outside ) is likewise pointwise the uniform distribution where every vertex is mapped independently.

()

If , then and are mapped at least units apart. In particular, every embedded edge has length .

Lemma 2.2.

With high probability, each unit ball meets simplices of .

Proof.

By an argument of Gromov and Guth, the probability that a random meets a fixed -simplex is .

Therefore, the expected number of simplices hitting is . If each simplex hitting was an independent event, then the probability that simplices meet would be ; therefore, with high probability, for every the number of simplices hitting would be . Indeed, complete independence is not necessary for this; the condition () is sufficient.

This condition holds when the simplices have no common vertices. Therefore, we can finish with a coloring trick, as in Gromov–Guth. We color the simplices of so that any two simplices that share a vertex are different colors. This can be done with colors. With high probability, the number of simplices of each color meeting is . Since the number of colors is , we are done.

Now we decompose each simplex into finer simplices, using the family of edgewise subdivisions due to Edelsbrunner and Grayson Reference EG00. This is a family of subdivisions of the standard -simplex with parameter which has the following relevant properties:

All links of interior vertices are isometric, and all links of boundary vertices are isometric to part of the interior link.

The subdivided simplices fall into at most isometry classes; in particular, all edges have length .

When we apply the edgewise subdivision with parameter , with the appropriate linear distortion, to , we get an embedding of a subdivided complex such that all edges have length by () and all links have thickness and hence .

Now we use the following lemma of Gromov and Guth:

Lemma 2.3.

For every , there is a way to move the vertices of by such that the resulting embedding is -thick.

If we choose sufficiently small compared to the edge lengths of , then there is an such that however we move vertices by , the links will still be -thick. Since these edge lengths are uniformly bounded below, this completes the proof.

3. Thick smooth embeddings

We now use Theorem 2.1 to build thick smooth embeddings of manifolds of bounded geometry.

Theorem 3.1.

Let be a closed Riemannian -manifold with and volume . Then for every , there is a smooth embedding such that

is contained in a ball of radius .

For every unit vector ,

The reach of is greater than , that is, the extension of to the exponential map on the normal bundle of vectors of length is an embedding.

Proof.

We prove this by reducing it to Theorem 2.1. That is, first we build a simplicial complex which is bi-Lipschitz to , with a bi-Lipschitz constant depending only on . We apply Theorem 2.1 to this complex to obtain a PL embedding and then smooth it out, using the fact that PL embeddings in the Whitney range are always smoothable. The quantitative bound on the smoothing follows from the fact that the local behavior of the PL embedding comes from a compact parameter space, allowing us to choose from a compact parameter space of local smoothings.

Throughout this proof we write to mean . This is different from the usage in the section 2. The first step is achieved by the following result.

Theorem 3.2.

There is a simplicial complex with at most simplices meeting at each vertex and a homeomorphism which is -bi-Lipschitz for some when is equipped with the standard simplex-wise metric.

Proof.

We start by constructing an -net of points on for an appropriate . We do this greedily: once we have chosen , we choose so that it is outside . In the end we get a set of points such that the -balls around them are disjoint and the -balls cover .

Now, Reference BDG17, Theorem 3 in particular gives the following:

Lemma 3.3.

If is small enough, there is a perturbation of to and a simplicial complex with a bi-Lipschitz homeomorphism as well as the following properties:

Its vertices are .

It is equipped with the piecewise linear metric determined by edge lengths which are geodesic distances in .

Its simplices have “thickness” ; this is defined to be the ratio of the least altitude of a vertex above the opposite face to the longest edge length. In particular, since the edge lengths are , this means that each simplex is -bi-Lipschitz to a standard one.

This automatically gives a bi-Lipschitz map to with the standard simplex-wise metric. Moreover, since has sectional curvatures , we immediately get a uniform bound on the local combinatorics of .

After applying this result to get , we apply Theorem 2.1, finding an embedding of a subdivision of which is 1-thick, lands in an -ball for , has -thick links, and expands all intrinsic distances by . In other words, we get a PL embedding .

For the sake of uniformity, we expand the metric of by a factor of ; this makes the embedding locally uniformly bi-Lipschitz. That is, for any such that ,

This is the property of which we actually use to construct a smoothing.

As in the main part of the paper, we assume that additionally has controlled th covariant derivatives of its curvature tensor for every . This allows us, as in Lemma 6.1, to fix an atlas for , with the following properties:

(1)

The also cover .

(2)

is the disjoint union of sets each consisting of pairwise disjoint charts.

(3)

The charts are uniformly bi-Lipschitz, and the th derivatives of all transition maps between charts are uniformly bounded depending only on and .

Here and both depend only on and . We construct our smoothing first on , then extend to , and so on by induction.

At each step of the induction, we use the following form of the weak Whitney embedding theorem Reference Hir76, §2.2, Thm. 2.13: for , the set of smooth embeddings is -dense in the set of continuous maps. Moreover, the set of smooth maps which restrict to some specific smooth map on a closed codimension  submanifold is likewise dense in the set of such continuous maps Reference Hir76, §2.2, Ex. 4.

The strategy is as follows. Note that the space of -bi-Lipschitz maps up to translation is compact by the Arzelà–Ascoli theorem. At every stage we also have a -compact space of possible partial local smoothings. Then Whitney will allow us to choose an extension from a space of possibilities which is also -compact.

We now give a detailed account of the inductive step. Suppose that we have defined a partial smooth embedding , where is a compact codimension submanifold of with

Moreover, suppose that is -close to for some sufficiently small depending on and , and that for each , , the partially defined function is an element of a -compact moduli space of maps each from one of a finite set of subdomains of to .

Fix a fine cubical mesh in ; it should be fine enough that any transition function sends a distance of to at least four times the diagonal of the cubes. The purpose of this mesh is to provide a uniformly finite set of subsets on which maps may be defined. Then, again by Arzelà–Ascoli, for any set which is a union of cubes in this mesh, the space of potential transition maps satisfying the bounds on the covariant derivatives in all degrees is -compact.

Fix . By the above, , again restricted to the union of cubes on which it is fully defined (call this domain ), is also chosen from a -compact moduli space , whose elements are patched together from a bounded number of compositions of elements of with transition maps as above. Of course, consists of the unique map from the empty set.

Let be the -compact set of -bi-Lipschitz embeddings . Notice that the subset consisting of pairs whose distance is is compact; this contains the pair .

Fix a smooth embedding . We say that is -good for , for some , if:

The distance between and is , where is fixed.

The map interpolating between and via a bump function, only depending on , whose transition lies within the layer of cubes touching the boundary of , has reach . (Here, we simply delete all boundary cubes outside of from the domain. Thus at this step the domain of actually recedes slightly; this is the motivation for the condition Equation 1.)

For any fixed pair , these are both open conditions in , so there is an open set of good pairs . Moreover, since (by Whitney) we can always choose a which coincides on with a given element of , these sets cover . Therefore, we can take a finite subcover corresponding to a set of pairs . Taking a cover by compact subsets subordinate to this, we get a compact set of allowable extensions of elements of to ; together with the modified sets of allowable maps on previous ’s (cut back so as to be defined on a domain of cubes) this makes .

We choose an extension of from the set of allowable extensions above. Doing this for every completes the induction step, giving some bound on the local geometry and reach by the compactness argument. Moreover, if we pick small enough compared to , then the embedding outside stays far enough away from the embedding inside. Nevertheless, all of these bounds become worse with every stage of the induction.

At the end of the induction, we have a smooth embedding of . Every choice we made was from a compact set of local smoothings depending ultimately only on and , which in turn controlled various bi-Lipschitz and bounds. Thus the resulting submanifold has . For the same reason, (as a map from with its original metric) has all directional derivatives . Moreover, since we did not move very far from , points from disjoint simplices cannot have gotten too close to each other. This, together with the local conditions, shows that has an embedded normal bundle of radius . By expanding everything by some additional we achieve the bounds desired in the statement of the theorem.

Acknowledgments

This paper owes a lot to the ideas of Gromov, as the introduction makes amply clear. The last author would like to thank Steve Ferry for a collaboration that began this work. Essentially, the polynomial bound in the nonoriented case can be obtained by combining Reference FW13 with some of the embedding arguments in this paper. We also thank MSRI for its hospitality during a semester (long ago) when we began working toward the results reported here. The authors would also like to thank Alexander Nabutovsky and Vitali Kapovitch for pointing out simplifications to several proofs, and for many useful conversations. Finally, we would like to thank the anonymous referee for a large number of remarkably insightful suggestions and corrections.

The authors of the appendix would like to thank Larry Guth and Sasha Berdnikov for stimulating correspondence during the writing of this appendix, and two anonymous referees for helpful comments regarding the exposition.

Figures

Figure 1.

Connecting preimages of opposite orientations with tubes: the global picture. Note that the Lipschitz constant of a null-homotopy depends only on the thickness of the tubes; therefore, inefficiencies in routing only matter insofar as they force many tubes to bunch up in the same region.

\renewcommand{\arraystretch}{1} \setlength{\unitlength}{1.0pt} \begin{tikzpicture} \tikzstyle{backing}=[white,line width=5pt]\draw(0,0) ellipse (4 and 2); \node(p1) at (-2.9,0.5) {$+$}; \node(p2) at (-3.1,-0.5) {$+$}; \node(p3) at (-1.8,1) {$+$}; \node(p4) at (-2,0) {$+$}; \node(p5) at (-2.2,-1) {$+$}; \node(p6) at (-0.8,1) {$+$}; \node(p7) at (-1,0) {$+$}; \node(p8) at (-1.2,-1) {$+$}; \node(m1) at (3.1,0.5) {$-$}; \node(m2) at (2.9,-0.5) {$-$}; \node(m3) at (2.2,1) {$-$}; \node(m4) at (2,0) {$-$}; \node(m5) at (1.8,-1) {$-$}; \node(m6) at (1.2,1) {$-$}; \node(m7) at (1,0) {$-$}; \node(m8) at (0.8,-1) {$-$}; \draw[backing,rounded corners] (p6) -- +(0,0.7) -- +(2,0.7) -- (m6); \draw[very thick,rounded corners] (p6) -- +(0,0.7) -- +(2,0.7) -- (m6); \draw[backing,rounded corners] (p7) -- +(0,0.7) -- +(2,0.7) -- (m7); \draw[very thick,rounded corners] (p7) -- +(0,0.7) -- +(2,0.7) -- (m7); \draw[backing,rounded corners] (p8) -- +(0,0.7) -- +(2,0.7) -- (m8); \draw[very thick,rounded corners] (p8) -- +(0,0.7) -- +(2,0.7) -- (m8); \draw[backing,rounded corners] (0.2,2.7) -- +(0.1,0.5) -- +(0.1,-0.5) -- +(-0.48,-2.9) -- +(0,-2.9); \draw[very thick,rounded corners] (0.2,2.7) -- +(0.1,0.5) -- +(0.1,-0.5) -- +(-0.48,-2.9) -- +(0,-2.9); \draw[backing,rounded corners] (p3) -- +(0,1.5) -- +(4,1.5) -- (m3); \draw[very thick,rounded corners] (p3) -- +(0,1.5) -- +(4,1.5) -- (m3); \draw[backing,rounded corners] (p4) -- +(0,1.5) -- +(4,1.5) -- (m4); \draw[very thick,rounded corners] (p4) -- +(0,1.5) -- +(4,1.5) -- (m4); \draw[backing,rounded corners] (p5) -- +(0,1.5) -- +(4,1.5) -- (m5); \draw[very thick,rounded corners] (p5) -- +(0,1.5) -- +(4,1.5) -- (m5); \draw[backing,rounded corners] (p1) -- +(0,1.7) -- +(3,1.7) -- +(3.1,2.2) +(3.07,-0.7) -- +(3.27,-0.7) -- +(3.27,0.3) -- +(3.55,1.7) -- +(6,1.7) -- (m1); \draw[very thick,rounded corners] (p1) -- +(0,1.7) -- +(3,1.7) -- +(3.1,2.2) +(3.07,-0.7) -- +(3.27,-0.7) -- +(3.27,0.3) -- +(3.55,1.7) -- +(6,1.7) -- (m1); \draw[backing,rounded corners] (p2) -- +(0,2.4) -- +(6,2.4) -- (m2); \draw[very thick,rounded corners] (p2) -- +(0,2.4) -- +(6,2.4) -- (m2); \end{tikzpicture}
Figure 2.

Constructing a null-homotopy: the local picture.

Figure 2(a)

Degrees on 2-cells of prisms.

\renewcommand{\arraystretch}{1} \setlength{\unitlength}{1.0pt} \begin{tikzpicture}[scale=1.2] \draw(0,2.5) -- (.2,0) -- (1.4,1.3) -- cycle; \node(base) at (.65,1.25){$\langle\omega,q \rangle$}; \node(top) at (8.5,1.5){$0$}; \draw(0,2.5) -- (.2,0) -- (1.4,1.3) -- cycle; \foreach\x in {1.5,3,4.5,8} { \draw(\x,2.5) -- (\x+1.4,1.3) -- (\x+.2,0); \draw[dotted] (\x,2.5) -- (\x+.2,0); } \draw(0,2.5) -- (8,2.5) (.2,0) -- (8.2,0) (1.4,1.3) -- (9.4,1.3); \node(etc) at (6.8,1.8){\huge{$\cdots$}}; \node(etc2) at (7,.7){\huge{$\cdots$}}; \node(side1) at (1.4,2){ $\left\lfloor\frac{1}{CL}\langle\alpha,p_0\rangle\right\rfloor$}; \draw[->] (2.7,2.8) node[anchor=west]{ $\left\lfloor\frac{2}{CL}\langle\alpha,p_0\rangle\right\rfloor -\left\lfloor\frac{1}{CL}\langle\alpha,p_0\rangle\right\rfloor$} .. controls (2.3,2.8) and (2.4,2.3) .. (2.9,1.8); \draw[->] (3.2,-.4) node[anchor=west]{ $\langle\omega,q \rangle-\sum_{p \in\partial q} \left\lfloor\frac{2}{CL}\langle\alpha,p\rangle\right\rfloor$} .. controls (2,-.4) and (4.5,1.2) .. (3.8,1.2); \draw[->] (8.7,2.5) node[anchor=south]{edge $p_0$} -- (8.7,2); \end{tikzpicture}
Figure 2(b)

Connecting homeomorphic preimages of with tubes.

\renewcommand{\arraystretch}{1} \setlength{\unitlength}{1.0pt} \begin{tikzpicture}[scale=1.2] \draw(0,2.5) -- (.2,0) -- (1.4,1.3) -- cycle; \draw(0,2.5) -- (.2,0) -- (1.4,1.3) -- cycle; \foreach\x in {1.5,3,4.5,8} { \draw(\x,2.5) -- (\x+1.4,1.3) -- (\x+.2,0); \draw[dotted] (\x,2.5) -- (\x+.2,0); } \draw(0,2.5) -- (8,2.5) (.2,0) -- (8.2,0) (1.4,1.3) -- (9.4,1.3); \node(etc) at (6.8,1.8){\huge{$\cdots$}}; \node(etc2) at (7,.7){\huge{$\cdots$}}; \draw(.6,1.25) circle [x radius = .25, y radius = .3] node{$+$}; \draw(.6,1.55) .. controls (1.2,1.55) .. (1.14,2); \draw(.6,.95) .. controls (1.6,.95) .. (1.66,1.8); \draw(1.4,1.9) circle [x radius = .2, y radius = .3, rotate=45] node{$-$}; \draw(3,.7) circle [x radius = .2, y radius = .3, rotate=-45] node{$+$}; \draw(5.1,1.55) -- (3.6,1.55) .. controls (2.74,1.55) .. (2.74,.6); \draw(5.1,.95) -- (3.6,.95) .. controls (3.26,.95) .. (3.26,.8); \draw(3.6,1.25) circle [x radius = .25, y radius = .3]; \draw(5.1,1.25) circle [x radius = .25, y radius = .3] node{$-$}; \draw(2.4,2.15) circle [x radius = .2, y radius = .3, rotate=45] node{$-$}; \draw(2.14,.2) -- (2.14,2.3) (2.66,.45) -- (2.66,2.05); \draw(2.4,.35) circle [x radius = .2, y radius = .3, rotate=-45] node{$+$}; \end{tikzpicture}
Figure 3.

An illustration of the subcomplex for , and , .

\renewcommand{\arraystretch}{1} \setlength{\unitlength}{1.0pt} \begin{tikzpicture} \tikzstyle{chainfill}=[fill=gray!40]\tikzstyle{dott}=[circle,fill=black,scale=0.4]\foreach\x in {0,1,2,3} { \draw[dashed] (0,\x+1) -- (3,\x+1); \draw[dashed] (\x,1) -- (\x,4); \node[dott] (l\x) at (0,\x+1) {}; \node[dott] (r\x) at (3,\x+1) {}; } \node[dott] (b1) at (1,1) {}; \node[dott] (b2) at (2,1) {}; \node[dott] (t1) at (1,4) {}; \node[dott] (t2) at (2,4) {}; \node[dott] (m1) at (1,2) {}; \node[dott] (m2) at (1,3) {}; \draw[very thick] (m1) -- (r1) (m2) -- (r2); \newcommand{\oo}{7.7} \newcommand{\ya}{-0.9} \newcommand{\yb}{0.3} \newcommand{\xa}{0.5} \newcommand{\xb}{0.55} \newcommand{\za}{0} \newcommand{\zb}{0.8} \foreach\x in {0,1,2,3} { \draw(\oo+\x* \xa+3*\ya+3*\za,\x* \xb+3*\yb+3*\zb) -- (\oo+\x* \xa+3*\ya,\x* \xb+3*\yb) -- (\oo+\x* \xa,\x* \xb); } \draw(\oo,0) -- (\oo+\xa* 3,\xb* 3) -- (\oo+\xa* 3+\ya* 3,\xb* 3+\yb* 3) -- (\oo+\ya* 3,\yb* 3) -- cycle; \filldraw[chainfill,thick] (\oo+\xa+2*\ya,\xb+2*\yb) -- (\oo+3*\xa+2*\ya,3*\xb+2*\yb) -- (\oo+3*\xa+2*\ya+\za,3*\xb+2*\yb+\zb) -- (\oo+\xa+2*\ya+\za,\xb+2*\yb+\zb) -- cycle; \filldraw[chainfill,thick] (\oo+\xa+\ya,\xb+\yb) -- (\oo+3*\xa+\ya,3*\xb+\yb) -- (\oo+3*\xa+\ya+\za,3*\xb+\yb+\zb) -- (\oo+\xa+\ya+\za,\xb+\yb+\zb) -- cycle; \draw(\oo+\ya+\za,\zb+\yb) -- (\oo+\za,\zb) -- (\oo+\xa+\za,\xb+\zb); \filldraw[chainfill,thick] (\oo+\ya+\za,\zb+\yb) -- (\oo+\xa+\ya+\za,\zb+\xb+\yb) -- (\oo+\xa+\za,\xb+\zb) -- (\oo+\xa* 3+\za,\zb+\xb* 3) -- (\oo+\xa* 3+\ya* 3+\za,\zb+\xb* 3+\yb* 3) -- (\oo+\ya* 3+\za,\zb+\yb* 3) -- cycle; \filldraw[chainfill,thick] (\oo+\xa+2*\ya+\za,\xb+2*\yb+\zb) -- (\oo+3*\xa+2*\ya+\za,3*\xb+2*\yb+\zb) -- (\oo+3*\xa+2*\ya+2*\za,3*\xb+2*\yb+2*\zb) -- (\oo+\xa+2*\ya+2*\za,\xb+2*\yb+2*\zb) -- cycle; \filldraw[chainfill,thick] (\oo+\xa+\ya+\za,\xb+\yb+\zb) -- (\oo+3*\xa+\ya+\za,3*\xb+\yb+\zb) -- (\oo+3*\xa+\ya+2*\za,3*\xb+\yb+2*\zb) -- (\oo+\xa+\ya+2*\za,\xb+\yb+2*\zb) -- cycle; \draw(\oo+\ya+2*\za,2*\zb+\yb) -- (\oo+2*\za,2*\zb) -- (\oo+\xa+2*\za,\xb+2*\zb); \filldraw[chainfill,thick] (\oo+\ya+2*\za,2*\zb+\yb) -- (\oo+\xa+\ya+2*\za,2*\zb+\xb+\yb) -- (\oo+\xa+2*\za,\xb+2*\zb) -- (\oo+\xa* 3+2*\za,2*\zb+\xb* 3) -- (\oo+\xa* 3+\ya* 3+2*\za,2*\zb+\xb* 3+\yb* 3) -- (\oo+\ya* 3+2*\za,2*\zb+\yb* 3) -- cycle; \filldraw[chainfill,thick] (\oo+\xa+2*\ya+2*\za,\xb+2*\yb+2*\zb) -- (\oo+3*\xa+2*\ya+2*\za,3*\xb+2*\yb+2*\zb) -- (\oo+3*\xa+2*\ya+3*\za,3*\xb+2*\yb+3*\zb) -- (\oo+\xa+2*\ya+3*\za,\xb+2*\yb+3*\zb) -- cycle; \filldraw[chainfill,thick] (\oo+\xa+\ya+2*\za,\xb+\yb+2*\zb) -- (\oo+3*\xa+\ya+2*\za,3*\xb+\yb+2*\zb) -- (\oo+3*\xa+\ya+3*\za,3*\xb+\yb+3*\zb) -- (\oo+\xa+\ya+3*\za,\xb+\yb+3*\zb) -- cycle; \draw(\oo+\za* 3,\zb* 3) -- (\oo+\xa* 3+\za* 3,\zb* 3+\xb* 3) -- (\oo+\xa* 3+\ya* 3+\za* 3,\zb* 3+\xb* 3+\yb* 3) -- (\oo+\ya* 3+\za* 3,\zb* 3+\yb* 3) -- cycle; \foreach\x in {0,1,2,3} { \draw(\oo+\x* \xa+3*\ya+3*\za,\x* \xb+3*\yb+3*\zb) -- (\oo+\x* \xa+3*\za,\x* \xb+3*\zb) -- (\oo+\x* \xa,\x* \xb); } \draw(\oo+\xa+\ya+3*\za,\xb+\yb+3*\zb) -- (\oo+\ya+3*\za,\yb+3*\zb); \draw(\oo+\xa+2*\ya+3*\za,\xb+2*\yb+3*\zb) -- (\oo+2*\ya+3*\za,2*\yb+3*\zb); \draw(\oo+\ya+3*\za,\yb+3*\zb) -- (\oo+\ya,\yb) --(\oo+\xa+\ya,\xb+\yb); \draw(\oo+2*\ya+3*\za,2*\yb+3*\zb) -- (\oo+2*\ya,2*\yb) -- (\oo+\xa+2*\ya,\xb+2*\yb); \end{tikzpicture}
Figure 4.

Illustrations for the proof of Lemma 4.1

Figure 4(a)

The second-order homotopy .

\renewcommand{\arraystretch}{1} \setlength{\unitlength}{1.0pt} \begin{tikzpicture}[scale=0.9] \node(Y) at (0.5,4) {$Y$}; \draw[very thick] (2,-0.4) .. controls (-1,0) and (-1,1.8) .. (0.5,2) .. controls (2,2.2) and (-1,4.4) .. (2.5,4.2) .. controls (6,4) and (6,2.4) .. (4,2) .. controls (2,1.6) and (5,-0.8) .. cycle; \coordinate(a1) at (0.5,0.3); \coordinate(a2) at (1.5,2); \coordinate(a3) at (2.5,0.3); \coordinate(b1) at (2,3.9); \coordinate(b2) at (2.9,5.5); \coordinate(b3) at (3.8,3.9); \draw(a1) -- (a2) -- (a3) -- cycle; \draw(b1) -- (b2) -- (b3) -- cycle; \draw(a1) -- (b1) -- (b3) -- (a3) -- cycle; \node[above] at (b2) {$f$}; \node[left] at (b1) {$g$}; \node[left] at (barycentric cs:b1=0.5,b2=0.5) {$H$}; \node[right] at (barycentric cs:b3=0.2,b2=0.8) {bounded}; \node[right] at (barycentric cs:b3=0.5,b2=0.5) {Lipschitz}; \node[right] at (barycentric cs:b3=0.75,b2=0.25) {constant}; \node[right] at (b3) {$g^\prime$}; \draw(a2) -- (b2); \end{tikzpicture}
Figure 4(b)

Extending from to .

\renewcommand{\arraystretch}{1} \setlength{\unitlength}{1.0pt} \begin{tikzpicture}[scale=0.9] \tikzstyle{chainfill}=[fill=gray!25, fill opacity=0.7]\tikzstyle{lightchainfill}=[fill=gray!10, fill opacity=0.5]\tikzstyle{dott}=[circle,fill=black,scale=0.6]\coordinate(a1) at (0.5,0.3); \coordinate(a0) at (1.5,2); \coordinate(a2) at (2.5,0.3); \coordinate(b1) at (1.25,2.1); \coordinate(b0) at (2.2,3.75); \coordinate(b2) at (3.15,2.1); \coordinate(w0) at (barycentric cs:b0=-0.5,a0=1.5); \coordinate(w1) at (barycentric cs:b1=-0.5,a1=1.5); \coordinate(w2) at (barycentric cs:b2=-0.5,a2=1.5); \coordinate(z0) at (barycentric cs:b0=1.8,a0=-0.8); \coordinate(z1) at (barycentric cs:b1=1.8,a1=-0.8); \coordinate(z2) at (barycentric cs:b2=1.8,a2=-0.8); \filldraw[chainfill] (a1) -- (a0) -- (b0) -- (b1) -- cycle; \filldraw[chainfill] (b1) -- (b2) -- (b0) -- cycle; \filldraw[chainfill] (a1) -- (a2) -- (a0) -- cycle; \fill[lightchainfill] (a0) -- (a2) -- (b2) -- (b0) -- cycle; \node[dott] (A2) at (a2) {}; \node[dott] (B2) at (b2) {}; \foreach\t in {0.1,0.2,...,0.9} { \draw[gray!50] (A2) -- (barycentric cs:a0=\t,a1=1-\t) -- (barycentric cs:b0=\t,b1=1-\t) -- (B2); } \draw[very thick] (A2) -- (a0) -- (b0) -- (B2); \draw(a0) -- (a1) -- (A2); \draw(w0) -- (w1) -- (w2) -- cycle (z0) -- (z1) -- (z2) -- cycle; \draw(w0) -- (z0) (w1) -- (z1); \draw[dashed] (a2) -- (b2); \draw[-stealth,very thick] (2.2-0.3,3.75+0.2) -- node[anchor=east,align=center] {null-homotopy\\of $\cap q$ in $B_1$} (1.25-0.3,2.1+0.2); \draw[stealth-] (barycentric cs:b0=0.7,a0=0.3) -- +(1,0) node[anchor=west] {$q$}; \draw[stealth-] (barycentric cs:b0=0.4,b2=0.6) -- +(1,0) node[anchor=west] {$\cap q$}; \draw[stealth-] (barycentric cs:a0=0.45,b2=0.45,a2=0.1) -- +(1.5,0) node[anchor=west] {$\Cap q$}; \draw[stealth-] (barycentric cs:z0=0.5,z1=0.5) -- +(-0.7,0.7) node[anchor=south east] {$B_1$}; \end{tikzpicture}
Figure 5.

A sequence of -spheres in , with north poles spaced at distance .

\renewcommand{\arraystretch}{1} \setlength{\unitlength}{1.0pt} \begin{tikzpicture}[scale=1.3] \tikzstyle{dott}=[circle,fill=black,scale=0.4]\foreach\x in {0,0.2,...,0.8} { \draw(\x,0) circle (1+\x); \node[dott] (l\x) at (2*\x+1,0) {}; } \end{tikzpicture}

Mathematical Fragments

Theorem A.

If is an (oriented) closed smooth null-cobordant manifold of complexity , then it has a null-cobordism of complexity at most

Theorem B.

Let be an -dimensional finite complex, and let be a finite complex which is rationally equivalent to a product of simply connected Eilenberg–MacLane spaces through dimension . If are -Lipschitz homotopic maps, then there is a homotopy between them which is -Lipschitz as a map from to .

Proposition 2.1 (Quantitative simplicial approximation theorem).

For finite simplicial complexes and with piecewise linear metrics, there are constants and such that any -Lipschitz map has a -Lipschitz simplicial approximation via a -Lipschitz homotopy.

Lemma 3.1 ( coisoperimetry).

Let be a finite simplicial complex equipped with the standard metric, and let be the cubical or edgewise -regular subdivision of , and let . Then there is a constant such that for any simplicial coboundary there is an with such that .

Lemma 3.2.

Let be a finite simplicial complex equipped with the standard metric, and let be an -regular subdivision of . Then for any , there is a constant such that for any simplicial coboundary , there is an with such that .

Lemma 3.3.

There is a such that boundaries of simplicial volume bound chains of simplicial volume at most .

Lemma 3.4.

Let be a finite simplicial complex. Then the following are equivalent for any constant :

any boundary has a filling with

any coboundary is the coboundary of some with .

Lemma 3.6.

The cubical and edgewise -regular subdivisions of both admit -spanning trees which are at most -gnarled; the gnarledness is bounded independent of .

Lemma 3.7.

Let be cubulated by a grid of side length , and let . We refer to

cells, i.e., faces of the cubulation;

faces, i.e., subcomplexes corresponding to faces of the unit cube; and

boxes, i.e., subcomplexes which are products of subintervals.

Then there is a -subcomplex of with the following properties:

.

deformation retracts to .

Every -cell of is homologous rel to a chain in whose intersection with each -face is a box.

More generally, every -dimensional box in is homologous rel to a chain in whose intersection with each -face is a box.

Lemma 3.8 (Bounded lifting).

Let be a finite simplicity complex. Fix a cocycle and a -spanning tree of . Then if lifts to a cocycle , we can find such a lift with .

Lemma 3.9.

Let be a finite simplicial complex equipped with the standard metric, let be the cubical or edgewise -regular subdivision of , and let . Then there is a constant such that for any , such that takes integer values and is a coboundary over , there is an such that and .

Lemma 4.1.

Let be a pair of finite simplicial complexes such that is finite for . Then there is a constant with the following property. Let be an -dimensional simplicial complex, and let be a simplicial map which is homotopic to a map . Then there is a short homotopy of to a map which is homotopic to in , that is, a homotopy which is -Lipschitz under the standard metric on the product cell structure on .

Theorem 4.2.

Let be a finite -dimensional simplicial complex, and let be a finite simplicial complex which has an -connected map , for some . Then there are constants and such that any two homotopic -Lipschitz maps are -Lipschitz homotopic through -Lipschitz maps.

Corollary 4.3.

Let be a rationally -connected finite complex, and let be an -dimensional finite complex. Then if , then there are constants and such that homotopic -Lipschitz maps from to are -Lipschitz homotopic through -Lipschitz maps.

Lemma 6.1.

Suppose that is a compact orientable -dimensional manifold with . There exists a finite atlas with the following properties, expressed in terms of constants , , and depending only on , as well as a natural number for some universal constant .

Every map in is the exponential map from the Euclidean -ball of radius to which agrees with the orientation of . Since the injectivity radius of is at least and , this is well-defined. We write

Here, is a geodesic ball of of radius , and is the Euclidean ball of radius in .

can be written as the disjoint union of sets of charts such that any pair of charts from the same have disjoint image.

When we restrict all the maps in to , they still cover .

The pullback of the metric with respect to every is comparable to the Euclidean metric, that is,

for every , for every , and where is the pullback of at .

The first and second derivatives of all transition maps are bounded by .

Lemma 6.2.

There is a function from to such that:

is monotonically increasing with and .

for all .

for all .

For every , there is some such that

for all .

Proposition 6.3.

Suppose that is a compact orientable -dimensional Riemannian manifold with volume and bounded local geometry. Then there is some (in particular depends only on ) such that is diffeomorphic to a submanifold with the following properties:

lies in a ball of radius .

The smooth map sending to , the oriented tangent space of at , has Lipschitz constant .

has a normal tubular neighborhood of size .

Lemma 6.4.

Suppose that is an embedded submanifold of satisfying the conclusions of Proposition 6.3. Then there is an embedding into the round unit sphere such that

has a tubular neighborhood of width . Additionally, can be extended to a -Lipschitz diffeomorphism from this tubular neighborhood to a neighborhood of width of .

The map given by has Lipschitz constant . Here, is the map from to from Proposition 6.3.

Theorem A.

Every closed smooth null-cobordant manifold of complexity has a filling of complexity at most , where for every .

Theorem C (Corollary of Reference BDG17, Thm. 3).

Every Riemannian -manifold of bounded geometry and volume is -bi-Lipschitz to a simplicial complex with vertices with each vertex lying in at most simplices. In particular, every smooth -manifold of complexity has a triangulation with vertices and each vertex lying in at most simplices.

Theorem 2.1.

Suppose that is a -dimensional simplicical complex with vertices and each vertex lying in at most simplices. Suppose that . Then there are and and a subdivision of which embeds linearly into the -dimensional Euclidean ball of radius

with Gromov–Guth thickness and such that for any -simplex of , the induced embedding is -thick.

Equation (1)

References

Reference [Ama15]
H. Amann, Uniformly regular and singular Riemannian manifolds, Elliptic and parabolic equations, Springer Proc. Math. Stat., vol. 119, Springer, Cham, 2015, pp. 1–43, DOI 10.1007/978-3-319-12547-3_1. MR3375165,
Show rawAMSref \bib{Amann2015}{article}{ label={Ama15}, author={Amann, Herbert}, title={Uniformly regular and singular Riemannian manifolds}, conference={ title={Elliptic and parabolic equations}, }, book={ series={Springer Proc. Math. Stat.}, volume={119}, publisher={Springer, Cham}, }, date={2015}, pages={1--43}, review={\MR {3375165}}, doi={10.1007/978-3-319-12547-3\_1}, }
Reference [Bau77]
H. J. Baues, Obstruction theory on homotopy classification of maps, Lecture Notes in Mathematics, Vol. 628, Springer-Verlag, Berlin-New York, 1977. MR0467748,
Show rawAMSref \bib{Baues}{book}{ label={Bau77}, author={Baues, Hans J.}, title={Obstruction theory on homotopy classification of maps}, series={Lecture Notes in Mathematics, Vol. 628}, publisher={Springer-Verlag, Berlin-New York}, date={1977}, pages={xi+387}, isbn={3-540-08534-3}, review={\MR {0467748}}, }
Reference [BDG17]
J.-D. Boissonnat, R. Dyer, and A. Ghosh, Delaunay triangulation of manifolds, Found. Comput. Math. 18 (2018), no. 2, 399–431, DOI 10.1007/s10208-017-9344-1. MR3777784,
Show rawAMSref \bib{BDG}{article}{ label={BDG17}, author={Boissonnat, Jean-Daniel}, author={Dyer, Ramsay}, author={Ghosh, Arijit}, title={Delaunay triangulation of manifolds}, journal={Found. Comput. Math.}, volume={18}, date={2018}, number={2}, pages={399--431}, issn={1615-3375}, review={\MR {3777784}}, doi={10.1007/s10208-017-9344-1}, }
Reference [BH81]
S. Buoncristiano and D. Hacon, An elementary geometric proof of two theorems of Thom, Topology 20 (1981), no. 1, 97–99, DOI 10.1016/0040-9383(81)90016-1. MR592572,
Show rawAMSref \bib{BH}{article}{ label={BH81}, author={Buoncristiano, Sandro}, author={Hacon, Derek}, title={An elementary geometric proof of two theorems of Thom}, journal={Topology}, volume={20}, date={1981}, number={1}, pages={97--99}, issn={0040-9383}, review={\MR {592572}}, doi={10.1016/0040-9383(81)90016-1}, }
Reference [Cha18]
G. R. Chambers, F. Manin, and S. Weinberger, Quantitative nullhomotopy and rational homotopy type, Geom. Funct. Anal. 28 (2018), no. 3, 563–588, DOI 10.1007/s00039-018-0450-2. MR3816519,
Show rawAMSref \bib{MR3816519}{article}{ label={Cha18}, author={Chambers, Gregory R.}, author={Manin, Fedor}, author={Weinberger, Shmuel}, title={Quantitative nullhomotopy and rational homotopy type}, journal={Geom. Funct. Anal.}, volume={28}, date={2018}, number={3}, pages={563--588}, issn={1016-443X}, review={\MR {3816519}}, doi={10.1007/s00039-018-0450-2}, }
Reference [CG85]
J. Cheeger and M. Gromov, On the characteristic numbers of complete manifolds of bounded curvature and finite volume, Differential geometry and complex analysis, Springer, Berlin, 1985, pp. 115–154. MR780040,
Show rawAMSref \bib{CheeGr}{article}{ label={CG85}, author={Cheeger, Jeff}, author={Gromov, Mikhael}, title={On the characteristic numbers of complete manifolds of bounded curvature and finite volume}, conference={ title={Differential geometry and complex analysis}, }, book={ publisher={Springer, Berlin}, }, date={1985}, pages={115--154}, review={\MR {780040}}, }
Reference [Che70]
J. Cheeger, Finiteness theorems for Riemannian manifolds, Amer. J. Math. 92 (1970), 61–74, DOI 10.2307/2373498. MR0263092,
Show rawAMSref \bib{CheeFin}{article}{ label={Che70}, author={Cheeger, Jeff}, title={Finiteness theorems for Riemannian manifolds}, journal={Amer. J. Math.}, volume={92}, date={1970}, pages={61--74}, issn={0002-9327}, review={\MR {0263092}}, doi={10.2307/2373498}, }
Reference [CT08]
F. Costantino and D. Thurston, 3-manifolds efficiently bound 4-manifolds, J. Topol. 1 (2008), no. 3, 703–745, DOI 10.1112/jtopol/jtn017. MR2417451,
Show rawAMSref \bib{CoTh}{article}{ label={CT08}, author={Costantino, Francesco}, author={Thurston, Dylan}, title={3-manifolds efficiently bound 4-manifolds}, journal={J. Topol.}, volume={1}, date={2008}, number={3}, pages={703--745}, issn={1753-8416}, review={\MR {2417451}}, doi={10.1112/jtopol/jtn017}, }
Reference [DKM09]
A. M. Duval, C. J. Klivans, and J. L. Martin, Simplicial matrix-tree theorems, Trans. Amer. Math. Soc. 361 (2009), no. 11, 6073–6114, DOI 10.1090/S0002-9947-09-04898-3. MR2529925,
Show rawAMSref \bib{DKM}{article}{ label={DKM09}, author={Duval, Art M.}, author={Klivans, Caroline J.}, author={Martin, Jeremy L.}, title={Simplicial matrix-tree theorems}, journal={Trans. Amer. Math. Soc.}, volume={361}, date={2009}, number={11}, pages={6073--6114}, issn={0002-9947}, review={\MR {2529925}}, doi={10.1090/S0002-9947-09-04898-3}, }
Reference [DKM11]
A. M. Duval, C. J. Klivans, and J. L. Martin, Cellular spanning trees and Laplacians of cubical complexes, Adv. in Appl. Math. 46 (2011), no. 1-4, 247–274, DOI 10.1016/j.aam.2010.05.005. MR2794024,
Show rawAMSref \bib{DKM2}{article}{ label={DKM11}, author={Duval, Art M.}, author={Klivans, Caroline J.}, author={Martin, Jeremy L.}, title={Cellular spanning trees and Laplacians of cubical complexes}, journal={Adv. in Appl. Math.}, volume={46}, date={2011}, number={1-4}, pages={247--274}, issn={0196-8858}, review={\MR {2794024}}, doi={10.1016/j.aam.2010.05.005}, }
Reference [DSS16]
M. Disconzi, Y. Shao, and G. Simonett, Some remarks on uniformly regular Riemannian manifolds, Math. Nachr. 289 (2016), no. 2-3, 232–242, DOI 10.1002/mana.201400354. MR3458304,
Show rawAMSref \bib{DMS}{article}{ label={DSS16}, author={Disconzi, Marcelo}, author={Shao, Yuanzhen}, author={Simonett, Gieri}, title={Some remarks on uniformly regular Riemannian manifolds}, journal={Math. Nachr.}, volume={289}, date={2016}, number={2-3}, pages={232--242}, issn={0025-584X}, review={\MR {3458304}}, doi={10.1002/mana.201400354}, }
Reference [EG00]
H. Edelsbrunner and D. R. Grayson, Edgewise subdivision of a simplex, Discrete Comput. Geom. 24 (2000), no. 4, 707–719, DOI 10.1145/304893.304897. ACM Symposium on Computational Geometry (Miami, FL, 1999). MR1799608,
Show rawAMSref \bib{EdGr}{article}{ label={EG00}, author={Edelsbrunner, H.}, author={Grayson, D. R.}, title={Edgewise subdivision of a simplex}, note={ACM Symposium on Computational Geometry (Miami, FL, 1999)}, journal={Discrete Comput. Geom.}, volume={24}, date={2000}, number={4}, pages={707--719}, issn={0179-5376}, review={\MR {1799608}}, doi={10.1145/304893.304897}, }
Reference [Eic91]
J. Eichhorn, The boundedness of connection coefficients and their derivatives, Math. Nachr. 152 (1991), 145–158, DOI 10.1002/mana.19911520113. MR1121230,
Show rawAMSref \bib{Eich}{article}{ label={Eic91}, author={Eichhorn, J\"urgen}, title={The boundedness of connection coefficients and their derivatives}, journal={Math. Nachr.}, volume={152}, date={1991}, pages={145--158}, issn={0025-584X}, review={\MR {1121230}}, doi={10.1002/mana.19911520113}, }
Reference [EK16]
S. Evra and T. Kaufman, Bounded degree cosystolic expanders of every dimension, STOC’16—Proceedings of the 48th Annual ACM SIGACT Symposium on Theory of Computing, ACM, New York, 2016, pp. 36–48. MR3536553,
Show rawAMSref \bib{EvKa}{article}{ label={EK16}, author={Evra, Shai}, author={Kaufman, Tali}, title={Bounded degree cosystolic expanders of every dimension}, conference={ title={STOC'16---Proceedings of the 48th Annual ACM SIGACT Symposium on Theory of Computing}, }, book={ publisher={ACM, New York}, }, date={2016}, pages={36--48}, review={\MR {3536553}}, }
Reference [EPC92]
D. B. A. Epstein, J. W. Cannon, D. F. Holt, S. V. F. Levy, M. S. Paterson, and W. P. Thurston, Word processing in groups, Jones and Bartlett Publishers, Boston, MA, 1992. MR1161694,
Show rawAMSref \bib{ECHLP}{book}{ label={EPC{\etalchar {+}}92}, author={Epstein, David B. A.}, author={Cannon, James W.}, author={Holt, Derek F.}, author={Levy, Silvio V. F.}, author={Paterson, Michael S.}, author={Thurston, William P.}, title={Word processing in groups}, publisher={Jones and Bartlett Publishers, Boston, MA}, date={1992}, pages={xii+330}, isbn={0-86720-244-0}, review={\MR {1161694}}, }
Reference [FF60]
H. Federer and W. H. Fleming, Normal and integral currents, Ann. of Math. (2) 72 (1960), 458–520, DOI 10.2307/1970227. MR0123260,
Show rawAMSref \bib{FF}{article}{ label={FF60}, author={Federer, Herbert}, author={Fleming, Wendell H.}, title={Normal and integral currents}, journal={Ann. of Math. (2)}, volume={72}, date={1960}, pages={458--520}, issn={0003-486X}, review={\MR {0123260}}, doi={10.2307/1970227}, }
Reference [FHT12]
Y. Félix, S. Halperin, and J.-C. Thomas, Rational homotopy theory, Graduate Texts in Mathematics, vol. 205, Springer-Verlag, New York, 2001, DOI 10.1007/978-1-4613-0105-9. MR1802847,
Show rawAMSref \bib{FHT}{book}{ label={FHT12}, author={F\'elix, Yves}, author={Halperin, Stephen}, author={Thomas, Jean-Claude}, title={Rational homotopy theory}, series={Graduate Texts in Mathematics}, volume={205}, publisher={Springer-Verlag, New York}, date={2001}, pages={xxxiv+535}, isbn={0-387-95068-0}, review={\MR {1802847}}, doi={10.1007/978-1-4613-0105-9}, }
Reference [FW13]
S. Ferry and S. Weinberger, Quantitative algebraic topology and Lipschitz homotopy, Proc. Natl. Acad. Sci. USA 110 (2013), no. 48, 19246–19250, DOI 10.1073/pnas.1208041110. MR3153953,
Show rawAMSref \bib{FWPNAS}{article}{ label={FW13}, author={Ferry, Steve}, author={Weinberger, Shmuel}, title={Quantitative algebraic topology and Lipschitz homotopy}, journal={Proc. Natl. Acad. Sci. USA}, volume={110}, date={2013}, number={48}, pages={19246--19250}, issn={1091-6490}, review={\MR {3153953}}, doi={10.1073/pnas.1208041110}, }
Reference [GG12]
M. Gromov and L. Guth, Generalizations of the Kolmogorov–Barzdin embedding estimates, Duke Math. J. 161 (2012), no. 13, 2549–2603, DOI 10.1215/00127094-1812840. MR2988903,
Show rawAMSref \bib{GrGu}{article}{ label={GG12}, author={Gromov, Misha}, author={Guth, Larry}, title={Generalizations of the Kolmogorov--Barzdin embedding estimates}, journal={Duke Math. J.}, volume={161}, date={2012}, number={13}, pages={2549--2603}, issn={0012-7094}, review={\MR {2988903}}, doi={10.1215/00127094-1812840}, }
Reference [GM81]
P. A. Griffiths and J. W. Morgan, Rational homotopy theory and differential forms, Progress in Mathematics, vol. 16, Birkhäuser, Boston, Mass., 1981. MR641551,
Show rawAMSref \bib{GM}{book}{ label={GM81}, author={Griffiths, Phillip A.}, author={Morgan, John W.}, title={Rational homotopy theory and differential forms}, series={Progress in Mathematics}, volume={16}, publisher={Birkh\"auser, Boston, Mass.}, date={1981}, pages={xi+242}, isbn={3-7643-3041-4}, review={\MR {641551}}, }
Reference [Gro96]
M. Gromov, Positive curvature, macroscopic dimension, spectral gaps and higher signatures, Functional analysis on the eve of the 21st century, Vol. II (New Brunswick, NJ, 1993), Progr. Math., vol. 132, Birkhäuser Boston, Boston, MA, 1996, pp. 1–213, DOI 10.1007/s10107-010-0354-x. MR1389019,
Show rawAMSref \bib{GroPC}{article}{ label={Gro96}, author={Gromov, M.}, title={Positive curvature, macroscopic dimension, spectral gaps and higher signatures}, conference={ title={Functional analysis on the eve of the 21st century, Vol.\ II}, address={New Brunswick, NJ}, date={1993}, }, book={ series={Progr. Math.}, volume={132}, publisher={Birkh\"auser Boston, Boston, MA}, }, date={1996}, pages={1--213}, review={\MR {1389019}}, doi={10.1007/s10107-010-0354-x}, }
Reference [Gro98]
M. Gromov, Metric structures for Riemannian and non-Riemannian spaces, vol. 152, Birkhäuser Boston, 1998.
Reference [Gro99]
M. Gromov, Quantitative homotopy theory, Invited Talks on the Occasion of the 250th Anniversary of Princeton University (H. Rossi, ed.), Prospects in Mathematics, 1999, pp. 45–49.
Reference [Hat01]
A. Hatcher, Algebraic topology, Cambridge University Press, Cambridge, 2002. MR1867354,
Show rawAMSref \bib{Hatc}{book}{ label={Hat01}, author={Hatcher, Allen}, title={Algebraic topology}, publisher={Cambridge University Press, Cambridge}, date={2002}, pages={xii+544}, isbn={0-521-79160-X}, isbn={0-521-79540-0}, review={\MR {1867354}}, }
Reference [Hir76]
M. W. Hirsch, Differential topology, Springer-Verlag, New York–Heidelberg, 1976. Graduate Texts in Mathematics, No. 33. MR0448362,
Show rawAMSref \bib{Hirsch}{book}{ label={Hir76}, author={Hirsch, Morris W.}, title={Differential topology}, note={Graduate Texts in Mathematics, No. 33}, publisher={Springer-Verlag, New York--Heidelberg}, date={1976}, pages={x+221}, review={\MR {0448362}}, }
Reference [HKW77]
S. Hildebrandt, H. Kaul, and K.-O. Widman, An existence theorem for harmonic mappings of Riemannian manifolds, Acta Math. 138 (1977), no. 1-2, 1–16, DOI 10.1007/BF02392311. MR0433502,
Show rawAMSref \bib{HKW}{article}{ label={HKW77}, author={Hildebrandt, St\'efan}, author={Kaul, Helmut}, author={Widman, Kjell-Ove}, title={An existence theorem for harmonic mappings of Riemannian manifolds}, journal={Acta Math.}, volume={138}, date={1977}, number={1-2}, pages={1--16}, issn={0001-5962}, review={\MR {0433502}}, doi={10.1007/BF02392311}, }
Reference [Kal83]
G. Kalai, Enumeration of -acyclic simplicial complexes, Israel J. Math. 45 (1983), no. 4, 337–351, DOI 10.1007/BF02804017. MR720308,
Show rawAMSref \bib{Kalai}{article}{ label={Kal83}, author={Kalai, Gil}, title={Enumeration of $\mathbf {Q}$-acyclic simplicial complexes}, journal={Israel J. Math.}, volume={45}, date={1983}, number={4}, pages={337--351}, issn={0021-2172}, review={\MR {720308}}, doi={10.1007/BF02804017}, }
Reference [KK04]
S. Klaus and M. Kreck, A quick proof of the rational Hurewicz theorem and a computation of the rational homotopy groups of spheres, Math. Proc. Cambridge Philos. Soc. 136 (2004), no. 3, 617–623, DOI 10.1017/S0305004103007114. MR2055050,
Show rawAMSref \bib{KK}{article}{ label={KK04}, author={Klaus, Stephan}, author={Kreck, Matthias}, title={A quick proof of the rational Hurewicz theorem and a computation of the rational homotopy groups of spheres}, journal={Math. Proc. Cambridge Philos. Soc.}, volume={136}, date={2004}, number={3}, pages={617--623}, issn={0305-0041}, review={\MR {2055050}}, doi={10.1017/S0305004103007114}, }
Reference [MS74]
J. W. Milnor and J. D. Stasheff, Characteristic classes, Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1974. Annals of Mathematics Studies, No. 76. MR0440554,
Show rawAMSref \bib{milnor1974characteristic}{book}{ label={MS74}, author={Milnor, John W.}, author={Stasheff, James D.}, title={Characteristic classes}, note={Annals of Mathematics Studies, No. 76}, publisher={Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo}, date={1974}, pages={vii+331}, review={\MR {0440554}}, }
Reference [Pet84]
S. Peters, Cheeger’s finiteness theorem for diffeomorphism classes of Riemannian manifolds, J. Reine Angew. Math. 349 (1984), 77–82, DOI 10.1515/crll.1984.349.77. MR743966,
Show rawAMSref \bib{Pet}{article}{ label={Pet84}, author={Peters, Stefan}, title={Cheeger's finiteness theorem for diffeomorphism classes of Riemannian manifolds}, journal={J. Reine Angew. Math.}, volume={349}, date={1984}, pages={77--82}, issn={0075-4102}, review={\MR {743966}}, doi={10.1515/crll.1984.349.77}, }
Reference [RS72]
C. P. Rourke and B. J. Sanderson, Introduction to piecewise-linear topology, Springer-Verlag, New York-Heidelberg, 1972. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 69. MR0350744,
Show rawAMSref \bib{pl_topology}{book}{ label={RS72}, author={Rourke, C. P.}, author={Sanderson, B. J.}, title={Introduction to piecewise-linear topology}, note={Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 69}, publisher={Springer-Verlag, New York-Heidelberg}, date={1972}, pages={viii+123}, review={\MR {0350744}}, }
Reference [Sch01]
T. Schick, Manifolds with boundary and of bounded geometry, Math. Nachr. 223 (2001), 103–120, DOI 10.1002/1522-2616(200103)223:1<103::AID-MANA103>3.3.CO;2-J. MR1817852,
Show rawAMSref \bib{Schick}{article}{ label={Sch01}, author={Schick, Thomas}, title={Manifolds with boundary and of bounded geometry}, journal={Math. Nachr.}, volume={223}, date={2001}, pages={103--120}, issn={0025-584X}, review={\MR {1817852}}, doi={10.1002/1522-2616(200103)223:1<103::AID-MANA103>3.3.CO;2-J}, }
Reference [Spa81]
E. H. Spanier, Algebraic topology, Springer-Verlag, New York-Berlin, 1981. Corrected reprint. MR666554,
Show rawAMSref \bib{Spa}{book}{ label={Spa81}, author={Spanier, Edwin H.}, title={Algebraic topology}, note={Corrected reprint}, publisher={Springer-Verlag, New York-Berlin}, date={1981}, pages={xvi+528}, isbn={0-387-90646-0}, review={\MR {666554}}, }
Reference [Sul74]
D. Sullivan, Genetics of homotopy theory and the Adams conjecture, Ann. of Math. (2) 100 (1974), 1–79, DOI 10.2307/1970841. MR0442930,
Show rawAMSref \bib{SulGen}{article}{ label={Sul74}, author={Sullivan, Dennis}, title={Genetics of homotopy theory and the Adams conjecture}, journal={Ann. of Math. (2)}, volume={100}, date={1974}, pages={1--79}, issn={0003-486X}, review={\MR {0442930}}, doi={10.2307/1970841}, }

Article Information

MSC 2010
Primary: 53C23 (Global geometric and topological methods ; differential geometric analysis on metric spaces)
Secondary: 57R75 (- and -cobordism)
Author Information
Gregory R. Chambers
Department of Mathematics, Rice University, Houston, Texas 77005
gchambers@rice.edu
MathSciNet
Dominic Dotterrer
Department of Computer Science, Stanford University, Stanford, California 94305
dominicd@cs.stanford.edu
MathSciNet
Fedor Manin
Department of Mathematics, Ohio State University, Columbus, Ohio 43210
manin@math.toronto.edu
ORCID
MathSciNet
Shmuel Weinberger
Department of Mathematics, University of Chicago, Chicago, Illinois 60637
shmuel@math.uchicago.edu
MathSciNet
Contributor Information
Fedor Manin
Shmuel Weinberger
Additional Notes

The first author was partially supported by NSERC Postdoctoral Fellowship PDF-487617-2016.

The fourth author was partially supported by NSF grant DMS-1510178.

Journal Information
Journal of the American Mathematical Society, Volume 31, Issue 4, ISSN 1088-6834, published by the American Mathematical Society, Providence, Rhode Island.
Publication History
This article was received on , revised on , , , and published on .
Copyright Information
Copyright 2018 American Mathematical Society
Article References
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  • DOI 10.1090/jams/903
  • MathSciNet Review: 3836564
  • Show rawAMSref \bib{3836564}{article}{ author={Chambers, Gregory}, author={Dotterrer, Dominic}, author={Manin, Fedor}, author={Weinberger, Shmuel}, title={Quantitative null-cobordism}, journal={J. Amer. Math. Soc.}, volume={31}, number={4}, date={2018-10}, pages={1165-1203}, issn={0894-0347}, review={3836564}, doi={10.1090/jams/903}, }

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