Quantitative null-cobordism

Abstract

For a given null-cobordant Riemannian $n$-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 $n$. In the appendix the bound is improved to one that is $O(L^{1+\varepsilon })$ for every $\varepsilon >0$.

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

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 $1$ sphere in the unit $n$-disk ($n>4$) 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 $h$-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 Thom⁠¹ 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., $\mathbb{Z}$ or $\mathbb{Z}/2\mathbb{Z}$) 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 $1$ and whose sectional curvature is everywhere between $-1$ and $1$. 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 $M$ 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 $M$ is not closed, we require it to look like a collar $\partial M \times [0,1]$ 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 Che70, Pet84, Gro98, §8D.

The central question is as follows. Given a smooth (oriented) manifold $M^n$ of complexity $V$ which is null-cobordant, what is the least complexity of a null-cobordism? That is, if $W$ is an (oriented) compact Riemannian $(n+1)$-manifold of bounded local geometry, which bounds a manifold diffeomorphic to $M$, how small can the volume of $W$ be? Gromov has observed Gro96, §5$\frac{5}{7}$ II that tracing through the relevant mathematics would give a tower of exponentials of $V$ (of size around the dimension of the manifold minus $2$), but he has suggested 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 $M$ is an (oriented) closed smooth null-cobordant manifold of complexity $V$, then it has a null-cobordism of complexity at most

$$c_1(n) V^{c_2(n)}.$$

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-cobordism⁠² 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 $M$ 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.⁠³

³

A proof of the nonoriented cobordism theorem was given by 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 $\exp (n^2)$-degree polynomial.

✖

For manifolds embedded in the sphere, the Lipschitz constant of the Thom map is closely related to the complexity of the submanifold⁠⁴ 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 $t$ in a null-homotopy (its “thickness”). Gromov has suggested Gro99 that these frequently have a linear bound for maps of finite complexes into finite simply connected complexes.⁠⁵

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 width⁠⁶ of the null-homotopy as well. This is a nontrivial issue: a null-homotopy of thickness $L$ can in general be replaced by one of width $\exp (L^d)$ where $d$ 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 $S^N$, we are forced to take $N$ 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 $X$ be an $n$-dimensional finite complex, and let $Y$ be a finite complex which is rationally equivalent to a product of simply connected Eilenberg–MacLane spaces through dimension $n$. If $f,g:X \rightarrow Y$ are $L$-Lipschitz homotopic maps, then there is a homotopy between them which is $C(X,Y)L$-Lipschitz as a map from $X \times [0,1]$ to $Y$.

The simplest settings in which this theorem applies are those in which $Y$ is an odd-dimensional sphere or in which $Y$ is a $2k$-sphere and $n \leq 4k-2$. More generally, $Y$ 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 $Y$.)

One antecedent to this result is given in 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 $X$. 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 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 $f: S^2 \rightarrow S^2$ is a degree $0$ map with Lipschitz constant $L$, then there is a $CL$-Lipschitz null-homotopy for some $C$.

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 $f:S^2 \to S^2$ be a null-homotopic $L$-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 $X$ is a subdivision of a tetrahedron into grid isometric subsimplices, $L$ to a side. The map $f$ 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 $S^2$ minus the basepoint, with degree $\pm 1$.

To construct a null-homotopy of $f$, we need to connect the positive and negative preimages with tubes in $X \times [0,1]$. 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 $X \times [0,1]$ and then connect them up in any available way.

To make this precise, assume that the tubes miss $X^{(0)} \times [0,1]$. Then we can count the number of tubes going through $p \times [0,1]$ for each 1-simplex $p$ of $X$. Every tube that goes into $q \times [0,1]$, for any 2-simplex $q$, must either come out through another edge or come back to 0. In other words, if $\alpha \in C^1(X;\mathbb{Z})$ is the cochain which indicates the number of tubes (with sign!) going through $p \times [0,1]$, then $\omega =\delta \alpha$ gives the degree of $f$ on 2-simplices of $X$. In the language of obstruction theory, $\omega$ is the obstruction to null-homotoping $f$, and the existence of $\alpha$ demonstrates that the obstruction can be resolved.

To ensure that it can be resolved efficiently, we need to pick a relatively small $\alpha$. The best we can do is to choose an $\alpha$ which takes values $\leq CL$. By considering a situation with degree $O(L^2)$ on one side of $X$ canceling out degree $-O(L^2)$ 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 $\alpha$, deciding how many tubes must go through a given point, we can connect them up in an entirely local way. We give $X \times [0,1]$ a cellulation by prisms of length $1/CL$ and base the 2-simplices of $X$. We then construct the map $F$ 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 $-3$ and $3$ in such a way that the map on the boundary of each prism has total degree $0$, 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 $\alpha$. If we ensure that for each 1-simplex $p$ of $X$, the degree of $F$ on $p \times [0,1]$ is $\langle \alpha ,p\rangle$, then $F$ will have degree 0 on the boundary of each “long prism” $q \times [0,1]$, where $q$ is a 2-simplex of $X$.

It remains to make sure that the degree is $0$ on the “short prisms”. To do this, we spread $\langle \alpha ,p\rangle$ as evenly as possible along the unit interval: for every integer $1 \leq t \leq CL$, the degree of $\alpha$ on $p \times [0,t/CL]$ is $\lfloor \frac{t}{CL}\langle \alpha ,p\rangle \rfloor$. This then also determines the required degree on $q \times \{t/CL\}$ for every 2-simplex $q$ and time $t$ to make the total degree on the boundary of each prism $0$. 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 $X \to Y$ between metric simplicial complexes, the fineness of the subdivision of $X$ 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 $X \times I$ 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 $X$ and $Y$. 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 $f$. 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 $1/L$ for a specified $L$. We also show that, for any simplicial map $f: X \rightarrow Y$ and any $L$, we can subdivide $X$ as above to form $X_L$ and homotope $f$ through a short homotopy to $\widetilde{f}: X_L \rightarrow Y$.

2.1. Regular subdivision of simplices

Definition.

Define a simplicial subdivision scheme to be a family, for every $n$ and $L$, of metric simplicial complexes $\Delta ^n(L)$ isometric to the standard $\Delta ^n$ with length 1 edges, such that $\Delta ^n(L)$ restricts to $\Delta ^{n-1}(L)$ on all faces. A subdivision scheme is regular if for each $n$ there is a constant $A_n$ such that $\Delta ^n(L)$ has at most $A_n$ isometry classes of simplices and a constant $r_n$ such that all 1-simplices of $\Delta ^n(L)$ have length in $[r_n^{-1}L^{-1},r_nL^{-1}]$.

Given a regular subdivision scheme, we can define the $L$-regular subdivision of any metric simplicial complex, where each simplex is replaced by an appropriately scaled copy of $\Delta ^n(L)$.

Note that $L$ 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 EG00, which has the advantages that the $L$-regular subdivision of $\Delta ^n(M)$ is $\Delta ^n(LM)$ and that the lengths of edges vary by a factor of only $\sqrt {2}$. Roughly, the method is to cut the simplex into small polyhedra by planes parallel to the $(n-1)$-dimensional faces, then partition each such polyhedron into simplices in a standard way. The other is described by Ferry and Weinberger FW13: the trick is to subdivide $\Delta ^n$ into $n+1$ identical cubes, then subdivide these in the obvious way into $L^n$ 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 $X$ and $Y$ with piecewise linear metrics, there are constants $C$ and $C^\prime$ such that any $L$-Lipschitz map $f:X \to Y$ has a $CL$-Lipschitz simplicial approximation via a $(CL+C^\prime )$-Lipschitz homotopy.

Proof.

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

Denote the open star of a vertex $v$ by $\operatorname {st}v$. Let $c$ be a Lebesgue number for the open cover $\{\operatorname {st}w \mid w$ is a vertex of $Y\}$ of $Y$, that is, a number such that every $c$-ball in $Y$ is contained in one of the sets in the cover. Then $c/L$ is a Lebesgue number for the open cover $\{f^{-1}(\operatorname {st}w)\}$ of $X$. Take a regular subdivision $X_L$ of $X$ so that for some $0<d(X)<1/2$ each simplex of $X_L$ has diameter between $dc/L$ and $c/2L$. Hence $f$ maps the closed star of each vertex $v$ of $X_L$ to the open star of some vertex $g(v)$ of $Y$. This gives us a map $g:X_L^{(0)} \to Y^{(0)}$ which takes adjacent vertices of $X_L$ to adjacent vertices of $Y$, and hence if $\ell$ is the maximum edge length of $Y$, $g$ is $\ell L/dc$-Lipschitz.

By a standard argument, this map $g$ extends linearly to a map $g:X_L \to Y$ with the same Lipschitz constant. The linear homotopy from $f$ to $g$ has Lipschitz constant $\max \{\ell L/dc, \ell \}$.

■

Remark.

Suppose that $Y$ and $X$ are $n$-dimensional and made up of standard simplices of edge length 1. Then $c=\frac{1}{\sqrt {2n(n+1)}}$ is the inradius of a standard simplex, and by using the edgewise subdivision, we can make sure that $d>1/2\sqrt {2}$. Thus the Lipschitz constant of the map increases by a factor of at most

$$C \leq 4\sqrt {n(n+1)}.$$

Furthermore, if $X$ is two-dimensional, then all of the edge lengths of the subdivision are equal. Therefore, in this case, $C \leq 4\sqrt {3}$ and, in fact, it approaches $2\sqrt {3}$ for large $L$, since we can choose a subdivision parameter very close to $L$ and, thus, $d$ very close to $1$.

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 ($\ell ^\infty$ coisoperimetry).

Let $X$ be a finite simplicial complex equipped with the standard metric, and let $X_L$ be the cubical or edgewise $L$-regular subdivision of $X$, and let $k \geq 1$. Then there is a constant $C_{\mathrm{IP}}=C_{\mathrm{IP}}(X,k)$ such that for any simplicial coboundary $\omega \in C^k(X_L;\mathbb{Z})$ there is an $\alpha \in C^{k-1}(X_L;\mathbb{Z})$ with $d\alpha =\omega$ such that $\lVert \alpha \rVert _\infty \leq C_{\mathrm{IP}}L\lVert \omega \rVert _\infty$.

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

Lemma 3.2.

Let $X$ be a finite simplicial complex equipped with the standard metric, and let $X_L$ be an $L$-regular subdivision of $X$. Then for any $k$, there is a constant $K=K(X,k)$ such that for any simplicial coboundary $\omega \in C^k(X_L;\mathbb{F})$, there is an $\alpha \in C^{k-1}(X_L;\mathbb{F})$ with $d\alpha =\omega$ such that $\lVert \alpha \rVert _\infty \leq KL\lVert \omega \rVert _\infty$.

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 $K=K(X,k)$ such that boundaries $b \in C_{k-1}(X_L;\mathbb{F})$ of simplicial volume $V$ bound chains of simplicial volume at most $KLV$.

Proof.

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

$$\operatorname {vol}\left(\sum \alpha _ip_i\right)=\sum \lvert \alpha _i\rvert .$$

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

$$\operatorname {mass}\left(\sum \alpha _ip_i\right)=\sum \lvert \alpha _i\rvert \operatorname {vol}_i(p_i).$$

Thus there are constants $K_i$ and $K^\prime _i$, depending on the choice of subdivision scheme, such that for every $i$-chain $c$,

$$K_iL^i\operatorname {mass}c \leq \operatorname {vol}c \leq K^\prime _iL^i.$$

Therefore to prove the lemma it suffices to show that a boundary whose $(k-1)$-mass in $X$ is $V$ bounds a chain whose $k$-mass is at most $KV$.

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 $n$ and $k+1$, we eventually end up with a shadow in the $k$-skeleton, which is the desired simplicial chain. Federer and Fleming’s original version FF60, Thm. 5.5 was based on deformation to the standard cubical lattice in $\mathbb{R}^n$. However, everything in their proof, except for the precise constants, translates to simplicial complexes. (See EPC$^{+}$92, 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 $W$ be an $n$-dimensional simplicial complex with the standard metric on each simplex. There is a constant $\rho (k,n)$ such that the following holds. Let $T$ be a Lipschitz $k$-chain in $W$ with coefficients in $\mathbb{F}$. Then we can write $T=P+Q+\partial S$, where

$(1)$

$\operatorname {mass}(P) \leq \rho (k,n)(\operatorname {mass}(T)+\operatorname {mass}(\partial T));$

$(2)$

$\operatorname {mass}(Q) \leq \rho (k,n)\operatorname {mass}(\partial T);$

$(3)$

$\operatorname {mass}(S) \leq \rho (k,n)\operatorname {mass}(T);$

$(4)$

$P$ can be expressed as an $\mathbb{F}$-linear combination of $k$-simplices of $W;$

$(5)$

if $\partial T$ can already be expressed as a combination of $(k-1)$-simplices of $W$ (for example, if $T$ is a cycle), then $Q=0$ and$$\operatorname {mass}(P) \leq \rho (k,n)(\operatorname {mass}(T).$$

Now suppose that $W$ is given a metric $d_W$ whose simplices are not standard, but such that the identity map $\iota :(W,d_{\mathrm{std}}) \to (W,d_W)$ satisfies

$$\lambda _1d(x,y) \leq d(\iota (x),\iota (y)) \leq \lambda _2d(x,y)$$

for all $x,y \in W$. When mass is measured with respect to $d_W$, the bounds in the theorem become

(1)

$\displaystyle {\operatorname {mass}(P) \leq \rho (k,n)\left(\frac{\lambda _2^k}{\lambda _1^k}\operatorname {mass}(T) +\frac{\lambda _2^{k-1}}{\lambda _1^k}\operatorname {mass}(\partial T)\right)}$;

(2)

$\displaystyle {\operatorname {mass}(Q) \leq \frac{\lambda _2^{k-1}}{\lambda _1^k}\rho (k,n)\operatorname {mass}(\partial T)}$;

(3)

$\displaystyle {\operatorname {mass}(S) \leq \frac{\lambda _2^{k+1}}{\lambda _1^k}\rho (k,n)\operatorname {mass}(T)}$.

We apply the theorem twice. First, we apply it to $b$ as a Lipschitz cycle in $X$ to show that it is homologous to a $(k-1)$-cycle $P \in C_{k-1}(X;\mathbb{F})$ of volume $\leq C(k)V$ via a Lipschitz $k$-chain $S$ of volume $\leq C(k)V$. Next, we apply it to $S$ as a $k$-chain in $X_L$. Notice that the ratio $\lambda _2/\lambda _1$ is bounded independent of $L$ for a regular subdivision; therefore, $S$ deforms rel boundary to a chain in $C_k(X_L;\mathbb{F})$ of volume $\leq C(k)C^\prime (k)V$, where $C^\prime$ depends on the subdivision scheme. Finally, $P$ bounds a chain in $X$ of volume $\leq C^\dagger C(k)V$, where $C^\dagger$ depends only on the geometry of $X$. Thus we can set $K=(C^\dagger +C^\prime )C$.

■

Lemma 3.4.

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

$(1)$

any boundary $\sigma \in C_{k-1}(X;\mathbb{F})$ has a filling $\tau \in C_k(X;\mathbb{F})$ with$$\operatorname {vol}\tau \leq C\operatorname {vol}\sigma ;$$

$(2)$

any coboundary $\omega \in C^k(X;\mathbb{F})$ is the coboundary of some $\alpha \in C^{k-1}(X;\mathbb{F})$ with $\lVert \alpha \rVert _\infty \leq C\lVert \omega \rVert _\infty$.

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 $L_\infty$-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 $T:(V,\lVert \cdot \rVert _V) \to (W,\lVert \cdot \rVert _W)$, and let $C(T)$ be the operator norm of the transformation

$$\bar{T}^{-1}: (\operatorname {im}T,\lVert \cdot \rVert _W) \to (V/\ker T,\lVert \cdot \rVert _{\bar{V}}),$$

where the norm of an equivalence class $\bar{v} \in V/\ker T$ is given by $\lVert \bar{v} \rVert _{\bar{V}}=\min _{v \in \bar{v}} \lVert v \rVert _V$. When $T$ is the boundary operator on $C_k(X;\mathbb{F})$, $C(T)$ is exactly the minimal constant $C$ in condition (1). Hence this is also the operator norm of the dual transformation $(\bar{T}^{-1})^*:\operatorname {im}T^* \to W^*/\ker T^*$. It remains to investigate the dual norms on these spaces.

By the Hahn–Banach theorem, any operator on $\operatorname {im}T$ extends to an operator of the same norm on all of $W$. Hence the dual norm of $\lVert \cdot \rVert _W|_{\operatorname {im}T}$ is exactly the norm $\lVert \overline{\varphi } \rVert _{\overline{W^*}}=\min _{\varphi \in \overline{\varphi }} \lVert \varphi \rVert _{W^*}$ on $W^*/\ker T^*$, and similarly the dual norm of $\lVert \cdot \rVert _{\bar{V}}$ is $\lVert \cdot \rVert _{V^*}|_{\operatorname {im}T^*}$. Therefore, the operator norm of $(\bar{T}^{-1})^*$ 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 $k$-spanning tree of a simplicial complex $X$ is a $k$-dimensional subcomplex $T$ which contains $X^{(k-1)}$, such that the induced map

$$H_{k-1}(T;\mathbb{Q}) \to H_{k-1}(X;\mathbb{Q})$$

is an isomorphism and $H_k(T;\mathbb{Q})=0$. A $k$-wrapping tree of $X$ is a $k$-dimensional subcomplex $U$ which contains $X^{(k-1)}$ and such that the induced maps

$$H_{k-1}(U;\mathbb{Q}) \to H_{k-1}(X;\mathbb{Q})\quad \text{and}\quad H_k(U;\mathbb{Q}) \to H_k(X;\mathbb{Q})$$

are both isomorphisms.

Lemma 3.5.

Every simplicial complex $X$ has a $k$-spanning tree and a $k$-wrapping tree.

Proof.

A $k$-spanning tree for any $X$ can be constucted greedily starting from $X^{(k-1)}$. At each step, we find a $k$-simplex $c$ in $X$ such that $\partial c$ represents a nonzero class in $H_{k-1}(T;\mathbb{Q})$ and add it to $T$. Once there are no such simplices left, $H_{k-1}(T;\mathbb{Q}) \to H_{k-1}(X;\mathbb{Q})$ is an isomorphism. By construction, $T$ has no rational $k$-cycles.

Notice that every $k$-simplex of $X$ outside $T$ is a cycle in $C_k(X,T;\mathbb{Q})$. To build a $k$-wrapping tree from a $k$-spanning tree, we may choose a basis for $H_k(X,T;\mathbb{Q})$ from among the simplices and add it to the tree.

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

Such spanning trees have been previously studied by Kalai Kal83 and Duval, Klivans, and Martin DKM09 and DKM11 in the case where $k$ is the dimension of the complex. In that case the $k$-simplices not contained in a spanning tree $T$ form a basis for $H_k(X,T;\mathbb{Q})$ (and a $k$-wrapping tree is simply the whole complex). When $X$ contains simplices in dimension $k+1$, however, there may be relations between the simplices when viewed as cycles in $X$ modulo $T$. The next definition attempts to quantify the extent to which such relations constrain the behavior of cocycles in the pair $(X,T)$.

Definition.

Let $T$ be a $k$-spanning tree of $X$. Consider the set $\mathcal{A}$ of vectors in $H_k(X,T;\mathbb{Q})$ which are images of $k$-simplices of $X$. We define the gnarledness

$$G(T)=\min \left\{\max _{a \in \mathcal{A}} \lVert a \rVert _1: \text{bases $\mathcal{B}$ for $H_k(X,T;\mathbb{Q})$ such that }\mathcal{A} \subset \mathbb{Z}\mathcal{B}\right\}.$$

We say that $T$ is $G(T)$-gnarled; we say a basis is optimal if $\max _{a \in \mathcal{A}} \lVert a \rVert _1$ 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 $2$ self-map of $S^1$,

$$X=S^1 \times [0,1]/(x,1) \sim (-x,1).$$

Let us say we take a one-dimensional spanning tree $T$ which includes all but one of the simplices of both $S^1 \times \{0\}$ and $S^1 \times \{1\}$; let $e_0$ and $e_1$, respectively, be the relevant 1-simplices in $X \setminus T$. Then in $H_1(X,T;\mathbb{Q}) \cong \mathbb{Q}$, $[e_0]=2$ and $[e_1]=1$. For any basis for $H_1(X,T;\mathbb{Q})$ in which $e_1$ is a lattice point, $\lVert e_0 \rVert _1 \geq 2$, so the tree $T$ is at least 2-gnarled. Indeed, the same will happen for any spanning tree of this complex.

Lemma 3.6.

The cubical and edgewise $L$-regular subdivisions of $X$ both admit $k$-spanning trees which are at most $C(X)$-gnarled; the gnarledness is bounded independent of $L$.

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 “$k$-spanning tree rel boundary” with good geometric properties. To be precise:

Lemma 3.7.

Let $K=I^n$ be cubulated by a grid of side length $1/r$, and let $k \leq n$. We refer to

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cells, i.e., faces of the cubulation;

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

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

Then there is a $k$-subcomplex $T \supset K^{(k-1)}$ of