Cobordism categories of manifolds with corners

In this paper we study the topology of cobordism categories of manifolds with corners. Specifically, if {Cob}_{d,} is the category whose objets are a fixed dimension d, with corners of codimension less than or equal to k, then we identify the homotopy type of the classifying space B{Cob}_{d,} as the zero space of a homotopy colimit of certain diagram of Thom spectra. We also identify the homotopy type of the corresponding cobordism category when extra tangential structure is assumed on the manifolds. These results generalize the results of Galatius, Madsen, Tillmann and Weiss, and the proofs are an adaptation of the their methods. As an application we describe the homotopy type of the category of open and closed strings with a background space X, as well as its higher dimensional analogues. This generalizes work of Baas-Cohen-Ramirez and Hanbury.


Introduction
Cobordism categories of manifolds with corners are of interest to both mathematicians and physicists. These categories are particularly relevant when studying open-closed topological and conformal field theories as such field theories are monoidal functors from these cobordism categories to categories such as vector spaces, chain complexes or other symmetric monoidal categories. Much work has already been done in this setting. See for example Moore [Moo01], Segal [Seg04], Costello [Cos07], Baas-Cohen-Ramirez [BCR06], and Hanbury [Han].
The main result of our paper, is a formula for calculating the homotopy type of the classifying space of the cobordism category Cob d, k of manifolds of fixed dimension d with corners of codimension ≤k. The result is the zero space of a homotopy colimit over a certain diagram of Thom spectra. More generally, let θ : B → BO(d) be a fibration. Then this recipe also allows us to compute the homotopy type of Cob θ d, k , the cobordism category whose morphisms are cobordisms with corners together with structure on the tangent bundle determined by θ. The precise definition of these categories will be given in section 4.
In some interesting cases we are able to evaluate this homotopy colimit explicitly. A fibration of particular interest is BSO(d)×X→BO(d), which is the composition The corresponding tangential structure is an orientation on the manifold, together with a map to a background space X. We will call this category Cob d, k (X) and we prove that BCob d, k (X) shares its homotopy type with X + ∧BCob d, k , a generalized homology theory of X. See the third and last part of section 6 for proof. The category Cob d, k (X) is particularly important in the 2-dimensional case where this category of "surfaces in X" plays a role in both Gromov-Witten theory as well as string topology. Variations on this category have been studied in several contexts, see Baas [Baa73], Sullivan [Sul71], Cohen-Madsen [CM], and Hanbury.
The kind of corners on manifolds we consider have been studied by Jänich [Jän68] and more recently by [Lau00]. The main feature of the corner structure under consideration is, as Laures showed, that such manifolds embed in R k + ×R N in such a way that the corner structure is preserved. Here R + is the nonnegative real numbers and R k + is the cartestian product (R + ) k . In turn, the corner structure induces a stratification of the underlying manifold by nested submanifolds whose inclusion maps into one another form a cubical diagram. In this way these manifolds with corners naturally live in the category of cubical diagrams of spaces which Laures referred to as k -spaces. Laures originally found cobordisms of k -manifolds while investigating Adams-Novikov resolutions, but k -manifolds have appeared in other contexts as well. They appear in open-closed string theory men-tioned above. R. Cohen, Jones, and Segal found that the space of gradient flow lines of a Morse function on a closed manifold M naturally has the structure of a framed manifold with corners [CJS95]. R. Cohen has used these ideas in his work on Floer theory [Cohar]. Lastly, k -manifolds naturally appear in the author's future work on stable automorphism groups of closed manifolds.
Our paper is also heavily indebted to the work of Galatius, Madsen, Tillmann, and Weiss [GMTW], who calculated the homotopy type of the cobordism category of closed manifolds as the zero space of a Thom spectrum. This paper is a generalization of their result and uses their techniques. From this perspective, the cost for considering manifolds with corners is that the classifying space of the cobordism category has the homotopy type of the zero space of a homotopy colimit over a cubical diagram of Thom spectra instead of a single spectrum.
The paper is organized as follows. In section 2 we discuss the cubical diagrams of spaces which, in the tradition of Laures's paper, we call k -spaces. In section 3 we proceed on to vector bundles in the k -space setting, and we introduce the analogue of prespectra in category of diagrams. In section 4 we define cobordism categories of manifolds with corners and state the main theorem of the paper. We prove the main theorem in sections 5. In section 6 we describe some applications including an explicit description of the homotopy type of Cob θ d, k as the zero space of well known spectra. Section 7 is an appendix of some results from differential topology and bundle theory modified to the k -manifold setting. We need these facts for the proof in section 5 and are used nowhere else.
The list of people the author would like to thank possibly stretches longer than this paper, so we restrict attention to three dear people. The author thanks David Ayala for many interesting conversations. The author thanks Soren Galatius for patiently explaining cobordism categories amongst other things and for his laconic, earnest support. Lastly, the author thanks Ralph Cohen, wise in things mathematical and otherwise. Without Prof. Cohen's sage guidance, this project would not exist.
The results of this paper are part of the author's Ph.D. thesis, written under the direction of Prof. Ralph Cohen at Stanford University.

k -manifolds and their embeddings
In this section we give an intrinsic definition of k -manifolds and discuss some of their basic properties. Most importantly, we explain what we mean by an embedding of a k -manifold and show that the space of embeddings of any k -manifold into (theorem 2.7). This generalizes the result of Laures which showed that the space of such embeddings is nonempty. Next we shall introduce k -spaces, which are (hyper)cubical diagrams of spaces. Just as smooth manifolds form a subcategory of topological spaces so too do k -manifolds naturally sit inside the category of k -spaces. In fact, all the usual objects in differential topology such as tangent bundles, normal bundles, embeddings, and Pontrjagin-Thom constructions have analogues in the setting of k -spaces.
As motivation, we describe a way to generate examples of k -manifolds. To this end, let a = (a 1 , ..., a k ) be a k-tuple of binary numbers and let R k (a) be the subspace of R k consisting of k-tuples (x 1 , ..., x k ) such that x i = 0 if a i = 0. Now suppose we have an embedded closed submanifold N → R k ×R n that is transverse to R k (a)×R n for all possible k-tuples a. As pointed out in [Lau00], the intersection of this submanifold with R k + ×R n is a manifold with corners. The k codimension one faces of this manifold are given by for a = (1, ..., 1, 0, 1, ..., 1). All of the higher codimension faces are of the same form but with other sequences a (it is entirely possible that these faces may be empty). Moreover the higher codimension strata fit together by inclusions induced by the inclusions of R k + (a)×R n into R k In [Lau00] some basic features of k -manifolds are observed: each of the submanifolds M (a) is an a -manifold and the product of an k -manifold with an l -manifold is a k + l -manifold. Now we give an intrinsic definition of k -manifolds as a special kind of manifolds with corners. The basic definition of a manifold with corners can be found in chapter 14 of J. Lee's book on manifolds, [Lee03]. Let U and V be open subsets of R k + . We say f : U →V is a diffeomorphism if f extends to an honest diffeomorphism F between two open sets in U , V ⊂R k such is a diffeomorphism in the sense described above.
Definition 2.1. A smooth structure with corners on a topological manifold M is a maximal collection of compatible charts with corners whose domains cover M . A topological manifold together with a smooth structure with corners is called a smooth manifold with corners.
As in [Lau00], for x = (x 1 , ..., x k ) ∈ R k + , c(x) is defined to be the number of x i 's that are zero. For m∈M , c(m) is defined as c(φ(x)). This number is independent of the chart φ. A face of M is a union of connected components of the space {m∈M | c(m) = 1 }.

Definition 2.2.
A k -manifold is a manifold M with corners together with an ordered k-tuple (∂ 1 M, ..., ∂ k M ) of subspaces satisfying the following properties: Examples of k -manifolds are closed manifolds ( 0 -manifolds), manifolds with boundary ( 1 -manifolds) and the unit k-dimensional cube (a k -manifold). k -manifolds have the feature that they admit the structure of a k + 1 -manifold. To do this, set ∂ k+1 M = ∅.
For more interesting examples, consider the the closed unit 3-ball that has a pinch around the equator (a 2 -manifold) and the tetrahedron which is a 4 -manifold (it is also easy to check the tetrahedron does not admit the structure of a 3 -manifold). The prototypical k -manifold is R k + . The compactification of R k + is a space of fundamental interest to us. To that end, define (R k + ) c := the one point compactification of R ∞ . It is easy to see that (R k + ) c is a k -manifold and the inclusion map is a map of k -manifolds. As a word of warning, it should be noted that ers do not admit the structure of a k -manifold. A simple example is the cardioid. Figure 1: The cardioid is a manifold with corners, but is not a k -manifold.
The reader has surely noticed the strata of a k -manifold and the inclusion maps between them form a (hyper)cubical diagram of spaces. Such diagrams will be called k -spaces and now we give them proper definitions and highlight their basic features.
Let 2 k be the poset whose objects are k-tuples of binary numbers. Given two elements a and b ∈ 2 k , a is less than b if when we write a = (a 1 , ..., a k ) In this instance we shall write a < b. Definition 2.3. A k -space is a functor from 2 k to Top, the category of topological spaces.
Definition 2.4. k -Top is the category whose objects are k -spaces, and whose morphisms are the natural transformations of these functors.
To obtain a k -space from a k -manifold M , set M ((1,...,1)) equal to M itself and for a =1 ∈ 2 k set The maps M (a < b) are the inclusion maps. We have just described a functor We continue on to describe some functors between the k -Top categories that are of fundamental importance. First, there is the product functor which extends the product functor for k -manifolds.
For 1 ≤ i ≤ k there are functors ∂ i , ∂ i : 2 k−1 → 2 k that insert a 0 and 1 respectively in between the i − 1 st and the i th slot of a ∈ 2 k−1 . Each of these functors induces a functor In the other direction, for 1≤i≤k there are functors I i : 2 k → 2 k−1 that forget the i th component of the vector a ∈ 2 k . Each of the functors I i induces a functor Finally, there is a functor which sends a space X to the constant k -space where every space X(a) is equal to X and every map X(a < b) is the identity.
Next we give an analogue of homotopy in the k -Top category. Let f 0 , f 1 : X→Y be k -maps. Consider [0, 1] as a 0 -space; thus X×I is also a k -space. After identifying X with X×{0} and X×{1} there are inclusion maps ι 0 , ι 1 : X→X×I respectively. Definition 2.5. A homotopy between k -maps f 0 and f 1 is a k map F : We should mention there are a number of conventions we hold for elements of 2 k . Provided it leads to no unambiguity, we use 0 as shorthand for (0, ..., 0), and 1 will stand for (1, ..., 1). We reserve e i to be the k-tuple whose components are all zero except for the ith. When a < b ∈ 2 k , we define b − a := (b 1 − a 1 , ..., b k − a k ) and for an element a ∈ 2 k , a is defined to be 1-a. Lastly, |a| denotes the number of non-zero component of a.
Now we discuss embeddings of k -manifolds.
Definition 2.6. A neat embedding of a k -manifold M is a natural transformation ι : , iii) these intersections are perpendicular, i.e. there is some > 0 such that We let Emb(M, R k + ×R N ) denote the space of all neat embeddings.
In [Lau00], Laures proved that every k -manifold neatly embeds in R k + ×R N for some N >> 0. This shows that any k -manifold is k -diffeomorphic to one of the examples constructed in the beginning of the chapter. Just like with closed manifolds, not only is the embedding space of a k -manifold nonempty, but for sufficiently large N it is weakly contractible. The main result of this section is Theorem 2.7. Emb(M, R k + ×R N ) is weakly contractible provided N is sufficiently large.
Proof. The theorem follows from a proposition which we now describe.
Let M be a k -manifold. Let A be a closed set in M (1) and let U be an open set in M (1) containing A. We obtain k -space structures on A and U by setting A(a) equal to A ∩ M (a) and U (a) equal to U ∩ M (a). Moreover, U is a k -manifold. Let e : U → R k + ×R N be a neat k -embedding. Finally, let us define Emb(M, R k + × R N , e) to be the set of neat k -embeddings of M that restrict to e on A.
Proposition 2.8. Emb(M, R k + ×R N , e) is nonempty for N sufficiently large. Proof of theorem 2.7 from proposition 2.8. Let φ represent an element of π n (Emb(M, R k + × R N )). The product of the adjoint of φ with the standard embedding of S n in R n+1 yields an embedding of k -manifolds, This map may be extended to a neat smooth embedding of k + 1 -manifolds Now apply proposition 2.8 to find a k + 1 -embedding It remains to prove proposition 2.8. . Call this neighborhood N , and now take a ∈ 2 k − B where without loss of generality we may assume that B contains all b strictly less than a. Pick a finite partition of unity for U , with the property that ∪U j is disjoint from A (∪ b∈B M (b)). If we define then e 1 is an embedding which agrees with e on a (possibly smaller) neigh- It also satisfies the hypothesis of lemma 2.9, setting this (possibly smaller) neighborhood equal to U in the notation of the lemma. But now by lemma 2.9 we may find a k -embedding of an open neighborhood of By construction, away from a neighborhood of e(U (e i )) the vector field agrees with E i , and on a smaller neighborhood of e(A(e i )), it agrees with V i . This extended vector field V i generates a flow which for small produces an embedding

Defineẽ
: By construction this map is well defined and is a k -embedding which extends the original embedding e in some neighborhood of A∪M (a). This concludes the proof of lemma 2.9 and hence also theorem 2.7.

k -vector bundles and k -spectra
In this section we explore vector bundles: (stable) tangent bundles, normal bundles, and structures over these bundles for k -manifolds. After describing this, we construct Thom spectra in the k -Top category analogous to ordinary Thom spectra. To keep indexing under control we will always use N to stand for d + n + k.
Definition 3.1. A k -vector bundle is a functor from 2 k to the category of vector bundles.
The k -vector bundles that appear in our paper belong to a special class of k -vector bundles.
The normal and tangent bundles of k -manifolds are geometric kvector bundles. As is the case with ordinary vector bundles, there is a universal example of geometric k -vector bundles which we now describe.
For the next definition, consider R as a 1 -space by letting R(0) = {0} and the map 0 < 1 be the inclusion of {0} into the real line.
Definition 3.3. The k -Grassmanian, which we denote by G(d, n) k , is the k -space defined as follows. For a ∈ 2 k let G(d, n)(a) be the space of d + |a| dimensional vector subspaces of R k (a)×R d+n . For a < b ∈ 2 k the map A neatly embedded d+k-dimensional k -submanifold of R k + ×R d+n admits a map to G(d,n) k by the rule We shall denote this k -map by Definition 3.4. The canonical geometric k -bundle which we denote by γ(d, n) k is a geometric k -vector bundle over G(d, n) k defined as follows. For a ∈ 2 k , the total space of γ(d, n)(a) is The reader is encouraged to notice that for every a ∈ 2 k , the bundle γ(d, n)(a) is canonically isomorphic to the usual canonical bundle γ(d+|a|, n).
Definition 3.5. The canonical perpendicular geometric k -bundle which we denote by γ ⊥ (d, n) k is a geometric k -vector bundle over G(d, n) k defined as follows. For a ∈ 2 k , the total space of γ(d, n)(a) is The reader is again encouraged to notice that for every a ∈ 2 k , the bundle γ ⊥ (d, n)(a) is canonically isomorphic to the usual canonical perpendicular bundle γ ⊥ (d+|a|, n).
At this juncture, we remark that the pullback of a vector bundle is readily defined for k -bundles. While the Thom space of a vector bundle doesn't generalize to all k -vector bundles, it does make sense for a large class of them.
Definition 3.6. Let ν : E k −→ B k be a k -vector bundle such that every vector bundle map E(a < b) restricts to an injective map on every fiber. Let D(E) k and S(E) k be the disk and sphere k -bundles associated to ν. The is induced by the map E(a < b).
Notice that the class of k -vector bundles which admit Thom spaces includes geometric k -vector bundles.
The next couple of remarks are more particular to our study of cobordisms accomodating stable tangential structure. In the early days of cobordism theory ( [Sto68]) it was realized that Pontrjagin-Thom construction could be generalized to yield a bijection between the set of cobordism classes of manifolds together with some stable structure and a homotopy group of the Thom space of the canonical perpendicular bundle over BO pulled back along a certain fibration. This fibration over BO corresponds to the extra stable tangential data. We briefly lay the groundwork for what the analogue of these fibrations will be in the k -space setting.
Definition 3.7. We say (B, σ , θ) is a structure if for each n≥0 we are given a k -space B d+n k together with maps of k -spaces We close out this section by discussing k -prespectra. The Thom spaces of the vector bundles sketched earlier will furnish us with important examples. For the rest of this section we will work solely with pointed spaces and pointed maps. Definition 3.9. A k -(pre)spectrum is a functor from 2 k to the category of (pre)spectra. Likewise, a k − Ω-spectrum is a functor from 2 k to the category of Ω-spectra.
It is the analogue of the sphere spectrum in the k -spectrum setting.
Definition 3.12. Let the Thom spectrum G −γ d k be the k -prespectrum defined as follows. For a ∈ 2 k , let G −γ d (a) be the Thom spectrum whose N th space is the Thom space of γ ⊥ (d, n)(a). The reader may notice this spectrum is isomorphic to the Thom spectrum Σ |a|−k M T (d + |a|). For a < b ∈ 2 k , let G −γ d (a < b) be the degree zero map of prespectra which on the N th space is simply the map of Thom spaces induced by the bundle map.
As an elaboration on the example above, suppose (B, σ , θ) is a structure.
Definition 3.13. We define the k -prespectrum B −θ * γ d k as follows. For a ∈ 2 k , let B −θ * γ d (a) be the Thom spectrum whose N th space is the Thom space of θ * γ ⊥ (d, n)(a). For a < b ∈ 2 k , let G −γ d (a < b) be the degree zero map of prespectra which on the N th space is simply the map of Thom spaces induced by the bundle map.
The indices can get confusing. For this reason, the reader may find it easier to view k -prespectra as a sequence of k -spaces together with maps of the suspension of each k -space to the following k -space. In the first of the two preceding definitions, and in the second definition, It is in order to note that the preceding definition is an example of a kdiagram of Thom spectra associated to virtual bundles. Next we describe a functor from k -prespectra to ordinary (not k !) Ω-spectra.
Definition 3.14. For any natural number n and pointed k -space X, let Ω n k X be the space of based k -maps from (R k Soon we will see that the zero spaces of Ω-spectra arising in this fashion are geometrically significant. Because of their centrality in the rest of the paper, we will abrreviate the zero space functor from k -prespectra to infinite loopspaces by just Ω ∞ k . An important proposition of Laures describes the homotopy type of Ω ∞ k X as being the zero space of an ordinary prespectrum. The spaces in the prespectrum are given by taking a homotopy colimit of an appropriate functor over the category 2 k * (notice the extra asterisk) which we now describe.
Following [Lau00], let 2 k * be the category consisting of 2 k plus an extra object, * , and one morphism from each object to * with the exception of 1 ∈ 2 k . If X is a functor from 2 k to the category of pointed spaces, then X * is the functor from 2 k * to pointed spaces defined by X * ( * ) = { basepoint}, and otherwise X * (a) := X(a) for all a ∈ 2 k . Now if X = {X N , σ N } is a k -prespectrum, the spaces hocolimX N * 2 k * form a prespectrum as there are maps Then At this point, we can state the analogue of the usual Pontrjagin-Thom theorem for k -manifolds.
More generally if we are given a structure (B, σ , θ) there is also the following theorem.
Proof. We prove the second more general statement. By the proposition above it suffices to show that there is a natural correspondance between cobordism classes and the set Composing C(a) with the compactification of the above map yields a map h(a) : The h(a) maps cobble together to yield a map H N −1 of k -spaces: This map is well defined up to k -homotopy.
To construct an inverse map we start with a map By Thom transversality for k -maps (see appendix 9) we may perturb will be a d − 1 + k dimensional k -submanifold of R k + ×R d−1+n ; notice the map H N −1 restricted to M is simply the classifying map of its tangent bundle, τ S, which we will now use as notation. Furthermore, we may also choose our homotopy l so that τ M is arbitrarily C 0 close to θ N −1 •H N −1 . Because of this, we may assume that M is a k -subspace of

k -Cobordism Categories and Statement of Main Theorem
The first goal of this section is to define cobordism categories for kmanifolds. The definition will be broad enough to include tangential data. Once we have done this, we state the main theorem of the paper.
Our cobordism categories are topological categories. We will first describe the category as a category of sets, and later indicate how to topologize the category. For the remainder of this section, let (B, σ , θ) be a structure over G(d, n) k . Also recall that for a k -submanifold M of R k + ×R ∞ , τ M : M −→ BO(d) k is the map classifying the tangent bundle of M . As a set, the objects of Cob θ d, k consist of triples (N, a, g) where: Morphisms of Cob θ d, k : The set of morphisms from (N 1 , a, g 1 ) to (N 2 , b, g 2 ) consist of quadruples (W, a, Here is an example when θ is trivial and d = k = 1. Notice that W (1, 0) is the disjoint union of figures (b) and (c).  Now we put a topology on the set of objects. As a preliminary step, notice that the space of k -maps, C k (X, Y ), can be written as an appropriate limit over a diagram where the spaces are of the form C(X(a), Y (b)) for a, b∈2 k and C(X(a), Y (b)) are endowed with the compact open topology. By definition then, Y (M ) n is the limit of the following diagram: which is topologized as a subset of We topologize the morphism space as we did the object space. Composition is cobordism composition and gluing the maps G along their shared domain. Now we are ready to state the main theorem of the paper. Again, recall that (B, σ , θ) is a structure over G(d, n) k .

Main Theorem 4.5.
BCob θ d, k The proof of this theorem will be the subject of the next section. Combining the theorem with the lemma from the previous section we obtain an important corollary which we will use for our applications.

Proof Of Main Theorem
The strategy of proof is to adapt the argument found in ( [GMTW]) to the setting of k -manifolds. We give an overview now; it should be noted that the original insights all come from [MW02]. First, there is a natural bijection between homotopy classes of maps, [X, Ω ∞−1 k B −θ * γ d k ], and the concordance classes of an associated sheaf of sets on X (our sheaves here are defined on manifolds without boundary). We describe this associated sheaf and explain what we mean by concordance. The proof of the bijection is postponed to the end of the section.
Definition 5.1. Let (B, σ , θ) be tangential data, let U be a manifold without boundary, and let N = d + n + k. We define D θ d,n, k (U ) to be the set of pairs (W, g) where W is a neat k -submanifold of U × R × R k + × R d−1+n and g : W −→ B d+n k is a k -map which satisfy the criteria listed below. For what follows, let π and f be the projection maps onto U and R respectively. Then ii) π is a submersion whose fibres are d+k-dimensional neat k -submanifolds of iii) θ N •g = T π W N , where T π W N is the map sending x ∈ W to the tangent space at x of the manifold π −1 (π(x)) thought of as a subspace of {x} × R k ×R d+n i.e. an element of G(d, n) k .

Proposition 5.3. Let X be a manifold without boundary. There is a natural bijection between
Proof. Postponed to the end of the section.
Definition 5.4. Let F be a sheaf of sets and X a manifold without boundary. Two sheaf elements s 0 , s 1 ∈ F(X) are concordant if there exists a sheaf element t ∈ F(X×R) such that i * 0 t = s 0 and i * 1 t = s 1 where i 0 and i 1 are the inclusion maps that identify X with X×{0} and X×{1} respectively. The set of concordance classes of elements of F(X) is denoted F[X].

So will we prove
. Contrast this with the following proposition. where ∆ l e = {(t 0 , ..., t k ) ∈ R l+1 | t i = 1} is the extended l-simplex.
Proof. Let X range over all spheres and invoke the two previous propositions.
We have seen that Ω ∞−1 k B −θ * γ d k is modelled by the realization of an appropriate sheaf. We might ask if something similar is true for Cob θ d, k . Indeed, a result from [MW02] is that there is a sheaf of categories C θ d, k which has the property that B|C θ d, k | is weak homotopy equivalent to BCob θ d, k . An analogous result in the setting of k -spaces is proved later in the section. From here it remains to show that B|C θ d, k | is equivalent to |D θ d, k |. We appear to be stuck since we are trying to show equivalence between a sheaf of categories and a sheaf of sets.
Fortunately, [MW02] (appendix A) provides us with two tools to show this equivalence. The first is a general procedure to take a sheaf of small categories F, and produce an associated sheaf of sets, βF such that there is a weak homotopy equivalence |βF| B|F|. Here is the recipe for constructing βF. Choose an uncountable indexing set J. An element of βF(X) is a pair (U, Ψ) where U = {U j |j∈J} is a locally finite open cover of X, and Ψ is a certain collection of morphisms. In detail: given a non-empty finite subset R ⊂ J, let Then Ψ is a collection ϕ R S ∈ N 1 F(U S ) indexed by pairs R ⊂ S of nonempty finite subsets of J subject to the conditions i) ϕ R R = id c R for an object c R ∈ N 0 F(U R ) ii)For each non-empty finite R ⊂ S, ϕ RS is a morphism from c S to c R |U S , iii) For all triples R ⊂ S ⊂ T of finite non-empty subsets of J, we have The second tool from [MW02] is a useful criteria for when a map between two sheaves of sets induces a weak homotopy equivalence on their realizations. It is called the relative surjectivity criterion. We give a description now. If A is a closed subset of X, and s ∈ colimF(U ) where U runs over open neighborhoods of A. Let F(X, a; s) denote the subset of F consisting of elements which agree with s in a neighborhood of A. Two elements t 0 , t 1 are concordant relative to A if they are concordant by a concordance whose germ near A is the constant concordance of s. Let F[X, A; s] denote the set of such concordance classes.
Proposition 5.8. [MW02] Relative Surjectivity Criteria -A map τ : F 1 → F 2 is a weak equivalence provided it induces a surjective map for all (X, A, s).
The rest of the section is devoted to the following tasks: 3) Finding a zig-zag of equivalent sheafs, using the Relative Surjectivity Criteria.
Definition 5.9. Let D θ,tr d, k be a sheaf of categories defined by letting D θ,tr d, k (U ) denote the set of triples (W, g, a) that satisfy: i) (W, g) ∈ D θ d, k (U ), ii) a : W −→ R is smooth, iii) for each x ∈ X, f : π −1 (x) −→ R is k -regular (see appendix 8.
Proposition 5.10. The forgetful map D θ,tr d, k −→ D θ d, k is a weak equivalence.
Proof. As stated in the introduction to this chapter, there is a known homotopy equivalence |βD θ,tr d, k | B|D θ,tr d, k | from [MW02], chapter 4.1 so it suffices to show the following lemma.
Proof. The proof follows exactly as in [GMTW] Proposition 4.2 provided we change "transverse" to " k -transverse". For the reader's sake, we include the argument here.
We must show the forgetful map βD θ,tr d, k → D θ d, k satisfies the relative surjectivity condition. To that end, let A be a closed subset of X, let (W, g) be an element of D . We can assume that q(x) > a R (x) for all R⊂J (make U smaller if not). Now for each a∈X − U , choose a∈R satisfying: Such an a exists by appendix 8 and furthermore the same value a will satisfy i) and ii) for all x in a small neighborhood U x ⊂X − A of X, so we can pick an open covering U = { U j | j∈J } of X − U , and real numbers a j such that i) and ii) are satisfied for all x∈U j . The covering U may be assumed to be locally finite. For each finite non-empty R⊂J , set a R = min{a j |j∈R}. For R⊂J(= J ∪ J ), write R = R ∪R with R ⊂J and R ⊂J , and define a R = a R if R =∅. This defines smooth functions a R : U R → R for all finite non-empty subsets R⊂J (a R is a constant function for R⊂J ) with the property that R ⊂ S implies a S ≤a R |U S . This defines an element of βD θ,tr d, k (X, A) which lifts W ∈D θ d, k (X) and extends the lift given near A.

This also ends the Proposition.
For what follows we will need the following convention. Suppose X is a smooth manifold without boundary and a 0 ,a 1 X→R are such that a 0 (x)≤a 1 (x) for all x∈X. Then X×(a 0 , a 1 ) := { (x, u) ∈ X×R | a 0 (x) < u < a 1 (x) }.
Definition 5.12. Given > 0 and two smooth functions a 0 , a 1 : X → R let C θ,tr d, k (X, a 0 , a 1 , ) be the set of pairs (W, g) where W is a neat k -submanifold of U ×(a 0 − , a 1 + )×R k + ×R d+n−1 and g : W →E N is a k -map which satisfy the criteria below. In what follows let π and f be the projection maps onto U and R respectively. i) π is a k -submersion with d + k dimensional fibers, ii) (π, f ) is proper, iii) (π, f ) : (π, f ) −1 (X×(a ν − , a ν + )) −→ X×(a ν − , a ν + ) is a ksubmersion for ν = 0, 1, iv) θ N •g = T π W N (see the definition of D θ d, k for the definition of T π W N ).
Definition 5.14. morC θ,tr d, k (X) := C θ,tr d, k (X, a 0 , a 1 ) ranging over all pairs (a 0 , a 1 ) of smooth functions where a 0 ≤ a 1 and the set of x ∈ X suc that a 0 (x) = a 1 (x) is a (possibly empty) union of connected components of X.
Two morphisms s ∈ C θ,tr d, k (X, a 0 , a 1 ) and t ∈ C θ,tr d, k (X, a 1 , a 2 ) are composable if there exists some : X→(0, ∞) and u ∈ C θ,tr d, k (X, a 1 − , a 1 + ) such that s = u ∈ C θ,tr d, k (a 1 − , a 1 ) and t = u ∈ C θ,tr d, k (a 1 , a 1 + ). In this case composition of the morphisms is given by taking the union of the two submanifolds and gluing the appropriate maps. The objects of C θ,tr d, k (X) are identified with the set of identity morphisms, C θ,tr d, k (X, a 0 , a 0 ).
Definition 5.15. Let C θ d, k (X) be the subcategory of C θ,tr d, k (X) consisting of morphisms (W, g, a, b) ∈ C θ,tr d, k (X, a, b) that satisfy two additional properties: Proposition 5.16. The inclusion functor C θ d, k (X) −→ C θ,tr d, k (X) is a weak equivalence.
Proof. The proof in [GMTW] (proposition 4.4 page 18) adapts easily to the k -space setting.
Proposition 5.17. There is an equivalence D θ,tr d, k −→ C θ,tr d, k . Proof. Again, the proof in [GMTW] (Proposition 4.3 page 18) adapts immediately to the k -space setting.
Proof. It suffices to show that for each p ≥ 0, the spaces N p Cob θ d, k and N p |C θ d, k | are weak homotopy equivalent. By a theorem of Milnor [Mil57], any space is weakly homotopic to the geometric realization of its singular set, so |. Now we argue that the singular set on the right is equivalent to a smooth singular set; we outline what we mean by smooth. Recall from definition 4.3 that Y (M ) is a subset of . For a smooth manifold X, we declare u : X −→ obCob θ d, k to be smooth if the projections onto and R are smooth. By smoothing theory, we see that C ∞ (∆ l e , N p Cob θ d, k ) is weakly homotopic to C(∆ l e , N p Cob θ d, k ) in the case p equals 0. The argument for higher p is identical. Thus Recall that the object space for Cob θ d, k is Now suppose X is a manifold without boundary and is a smooth map. Then ( graph(u 1 ) , g•π 2 ) is an element of obC θ d, k (X) (where π 2 is projection on the graph of u 1 ). Conversely, given an object (W, g) ∈ obC θ d, k (X) we obtain a smooth map X −→ obCob θ d, k by sending x ∈ X to ( π −1 (x) , g |π −1 (x) ). Thus we've shown that when p equals 0. The cases p > 0 are similar. Setting X = ∆ l e yields Informally, the reader should think of elements of D θ d, k as being parametrized d + k dimensional k -manifolds, W , along with a proper map to R (the "cobordism direction"). Going from right-to-left we convert these elements until they are parametrized morphisms of Cob θ d, k . To that end, in (5) we added in the extra information of a parametrized beginning and parametrized end slice which W meets transversally. Then we discarded the rest of the manifold outside the slices in (3), and finally with (2) we insisted that W meets the boundary slices not just transversely, but perpendicular to the R "cobordism" direction. To prove maps (2)-(5) are weak homotopy equivalences we made significant use of the Relative Surjectivity Criterion. That left the maps (1) and (6). The first map was loosely a Yoneda embedding. The last map is based on the Pontrjagin-Thom construction. That it is an equivalence is to be shown.
Proof. From proposition 5.5 it suffices to construct a bijection ρ and its inverse σ between relative concordance classes D θ d, k [X, A; s 0 ] and [X, A; Ω ∞−1 k B −θ * γ d k , s 0 ] for any closed manifold X. Thus, by allowing our choice of X to range over spheres of arbitrary dimension, and letting A be a point we obtain our result. The relative version follows along lines similar to the absolute case, which we now present.
We construct σ first. Let X be a closed manifold and let us pick a map h represeneting the class n) for some N > 0. Its adjoint map is a k -map. After removing the point at infinity from X + ∧(R k + ×R d+n−1 ) we are left a k -space which we identify with n)) is also a k -manifold. Also, h restricts to a map We are now in a situation where we may apply appendix 9. Thus, θ N h may be perturbed under a homotopy l to a smooth map h 0 of k -manifolds transverse to G(d, n) k . Set which is a codimension n k -submanifold of X×R k + ×R d+n−1 which up to concordance is a neat k -submanifold. Let π be the projection from V onto X and ι the inclusion map of V into X×R k + ×R d+n−1 . By construction, the normal bundle ν of V in X×R k Then we have the following k -bundle isomorphism: On the otherhand from the fact that V is a submanifold of X×R k + ×R d+n−1 we also have the k -bundle isomorphism: Piecing these two lines together gives an isomorphism By appendix 12, this isomorphism is induced up to k -homotopy by a unique k isomorphism let p : W → V be projection, and consider which is evidently a bundle surjection and induced by projection onto X. By the Philips Submersion Theorem (appendix 11), π * • Φ•p * is homotopic to a k -submersion s through some homotopy s t . Unfortunately, s is now no longer the same map as the projection of W onto X. To remedy this, pick an immersion e be of W into R n , and φ : I → I a smooth monotone increasing function that is zero in a neighborhood of zero and one in a neighborhood of one. Then is a k -isotopy from the identity embedding of W to an embedding id W ×e 1 where e 1 is also a k -embedding. Follow this isotopy by s t ×e 1 . Then is a k -submanifold of X×R×R d−1+n+n such that projection onto X is a submersion. The reader should notice that we may assume the projection f of W 0 onto the first R factor is proper.
To produce the map of tangential data g : W 0 →θ * N γ ⊥ (d, n) k . Since θ N is a fibration, it suffices to find a map g : W 0 →θ * N γ ⊥ (d, n) k and a homotopy from θ N g to T π W 0 . The map g is provided by the composition W 0 Recall that h is the map from the very beginning of our construction, and we have chosen a homotopy l from θ N h to h 0 . Also note that h 0 •p is precisely the map classifying the normal bundle of W in X×R×R k + ×R d+n−1 and homotopy s and the k -isotopy from the previous paragraph provide a homotopy from h 0 •p to T π W 0 to T π W 0 . Now we construct ρ. Suppose we are given (W, g) ∈ D θ d, k [X]. Recall that W is a k -submanifold of X×R×R k + ×R d+n−1 . By appendix 8 there is a regular value c ∈ R for the projection map f : W → R. Set M is a k -submanifold of X×{c}×R k + ×R d+n−1 . The normal bundle µ of M in X×{a}×R k + ×R d+n−1 is a k -vector bundle and for any a ∈ 2 k the Thom collapse map produces a map These maps assemble to form a k -map After picking a metric on X, the normal bundle ω of W ⊂X×R×R k + ×R d+n−1 fits in the following short exact sequence of vector bundles Since π * : T W → T X is a bundle surjection, and π is nothing more than π 1 •ι the sequence is isomorphic to This identifies the normal bundle ω as the total space of the bundle π 2 ι * γ ⊥ (d, n) k . The product of this map with g yields a k -map An argument identical to the one above shows that the normal bundle ω restricted to M is isomorphic to µ (replace X and π with R and f ). This yields the following sequence of maps The composition of this map induces a map on the Thom space Precomposing this map with the Thom collapse map and taking the adjoint with respect to X produces a map A check shows that ρ and σ are homotopy inverse to each other.

Applications
Unoriented and Oriented < k > -manifolds Proposition 6.1. BCob d, k Ω ∞−1 X where X is the following suspension spectrum: Let θ : G(d, n) + k →G(d, n) k be the structure map from the oriented k -Grassmanian to the k -Grassmanian which forgets orientation.
Proof. When k = 0, there is nothing to prove, since the homotopy colimit is over the trivial category with one object. When k = 1, the cofiber of is known to be Σ ∞ BO(d + 1) + , and the cofiber of It remains to evaluate the homotopy colimit when k ≥ 2 where the homotopy colimit is either hocolim For any k -space E k , the homotopy colimit hocolim 2 k * Gluing these triangles onto the previous squares yields a diagram commuting up to a prescribed homotopy: for all a < b ∈ ∂ k 2 k−1 . From this we see that M θ actually induces a map When θ = 0, M * 0 = G −γ d (a < a + e k ) and so F 0 = f . For θ = 1, after identifying G −γ d (a) with G −γ d (a ) where a = a − e k−1 + e k , we see that M * 1 = G −γ d (a < a + e k ) and so F 1 factors through hocolim Applying the lemma gives which after unraveling the definition of γ ⊥ (d, n) k we see is precisely By induction on k we have already identified the homotopy type of the righthand side. The result follows from repeated application of Pascal's Triangle identity.

-The restriction functors
Let 0 ≤ l < k, 0 ≤ d, and c ∈ 2 k with |c| = l. The functor 2 l → 2 k (c) → 2 k induces a restriction functor which sends (W, g, a, b) to (W (c), g(c), a, b). We focus on the case when k = 1 and l = 0. From the previous section we know that Ψ induces a map on classifying spaces Theorem 6.4. Let k = 1(l = 0) and θ be the trivial or orientation fibration. Then up to homotopy Ψ is induced by the spectrum map Σ ∞ BO(d + 1) −→ M T (d) which occurs in the cofibration sequence Σ −1 M T (d) → M T (d + 1) → Σ ∞ BO(d + 1) described in [GMTW]. When d = k = 1, this has been identified as the circle transfer.
BΨ is induced by the spectrum map which is the identity on the first summand and collapses the second summand to a point.
Proof. Notice the following diagram is a commutative diagram of topological categories: Here P is the functor which assigns to any topological space its path-space category. The horizontal maps are defined by sending a morphism (W, g, a, b) to the path which at time a ≤ t ≤ b is the composition of the Thom collapse of (R ∞ ×R k + ×{t}) c to the tubular neighborhood of the subspace of W lying over t with the map g * . On classifying spaces, this map induces the homotopy equivalence of 4.5. The right vertical map is simply restriction. Now we focus on the special case where θ is the identity, or the orientation map, k = 1 and l = 0. Then we have another diagram which commutes up to homotopy: The vertical map on the left is in the same homotopy class as the map on classifying spaces induced by the right vertical map in the previous square.
The horizontal top map is the equivalence from proposition 3.15, which in this case reduces to the fact that the homotopy fiber of a map between zero spaces of spectra is homotopic to the zerosection of the homotopy cofiber of the corresponding map. It follows that defining the right vertical map to be the map in the Puppe sequence yields a homotopy commutative diagram.
It is known ( [MS00]) that when d = k = 1, l = 0, and θ is trivial or an orientation, the map on the right is induced by a circle transfer. Thus in the oriented case, we have identified this map as being up to homotopy the map induced by the circle transfer For general 0 ≤ l < k, notice that any restriction functor is the composition of restriction functors where l = k − 1; thus it suffices to consider this special case. We have just dealt with the case k = 1. Now we investigate larger k and find the maps to be less interesting. The argument precedes as before, so we find a homotopy commutative diagram: We've already established the cofiber is of a nullhomotopic map, so the vertical map on the right is given by id ∨ * .

-Maps To A Background Space
Let X be any topological space and let (B, σ , θ) be tangential data. Then consider the category Cob θ d, k (X) whose objects are pairs (α, x) where α = (N, a, f ) is an object in Cob θ d, k and x is a map from N to X. Likewise its morphisms are pairs (β, x) where β = (W, a, b, F ) is a morphism in Cob θ d, k and x is a map from W to X.  We shall inductively extend this result to k -manifolds. The inductive hypothesis that seems to work naturally involves a broader class of kspaces than k -manifolds. Every k -manifold is a type 17 k -space. One important feature type 17 k -spaces possess that k -manifolds do not is that they are closed under the ∂ i operation. We will use this fact. now we expand the limit out = lim(lim (a,b)∈P k−1 ∂ k X(a, b)→lim (a,b)∈P k−1 X(a, b+e k )←lim (a,b)∈P k−1 ∂ k X(a, b)) The map on the right is induced by restriction to M (1-e k ) and is a fibration so we may replace the limit by a homtopy limit without affecting homotopy type.
holim(lim (a,b)∈P k−1 ∂ k X(a, b)→lim (a,b)∈P k−1 X(a, b+e k )←lim (a,b)∈P k−1 ∂ k X(a, b)) Now by induction on k, holim(lim (a,b)∈P k−1 ∂ k Y (a, b)→lim (a,b)∈P k−1 X(a, b+e k )→lim (a,b)∈P k−1 ∂ k Y (a, b)) The middle term may also be dealt with. Because the spaces in the middle limit are entirely of the form C ∞ (M (a), N (b)) and the map is a weak equivalence, by an inductive argument similar to the one we are in the process of giving here, we may conclude that lim (a,b)∈P k−1 X(a, b + e k ) lim (a,b)∈P k−1 Y (a, b + e k ) and hence holim(lim (a,b)∈P k−1 ∂ k Y (a, b)→lim (a,b)∈P k−1 Y (a, b+e k )←lim (a,b)∈P k−1 ∂ k Y (a, b)) lim(lim (a,b)∈P k−1 ∂ k Y (a, b)→lim (a,b)∈P k−1 Y (a, b+e k )←lim (a,b)∈P k−1 ∂ k Y (a, b)) = Sur k (T M, T N ).
The first arrow is a fibration so we may replace the outer lim with holim, or in this simple case, the homotopy pullback, and preserve the homotopy type.
= holim(lim P k−1 Γ a+e k (c + e k ) → lim P k−1 Γ a (c + e k ) ← lim P k−1 Γ a (c)) By induction, the stabilization map on lim P k−1 Γ a+e k (c+e k ) and lim P k−1 Γ a (c)) is d 1 and d 2 connected respectively, but lim P k−1 Γ a (c + e k ) is d 3 > d 1 connected since for every a∈2 k−1 we have dimRes a U (a + e k ) − M (a) > dimU (a + e k ) − M (a + e k ) Now we may use the mini-Lemma to conclude that the stabilization map on Γ k (U, V ) is min{d 1 , d 2 } connected, but min{d 1 , d 2 } = min a∈2 k d(a).