Bordism and Cobordism

Monday, July 23rd, 2012 | Author:

Two connected compact manifolds N and M are said to be bordant, if there exists a manifold W with boundary consisting of two connected components isomorphic to N and M respectively. The name comes from french and means sharing a boundary. Some people say cobordant, since the manifolds don't share a boundary but "are" shared as a boundary (I don't know how to explain this better than with the definition given above). We will stick to "bordant" because we investigate precisely what "the bordism of a manifold" and "the cobordism of a manifold" are.

One can see that being bordant is an equivalence relation, so it makes sense to speak of bordism classes of manifolds. By enriching N and M with extra structure (like a tangential framing, or an orientation), we get several different notions of bordism classes.

From each of these bordism theories, we get a sequence of spaces \Omega_n such that \Omega_n is the Thom space of a universal bundle over some classifying space (I will explain that later) and \Sigma \Omega_n is homotopy equivalent to \Omega_{n+1}. Homotopy theorists like to call such a sequence then a spectrum and by standard theory one can associate to each spectrum a generalized homology theory and a generalized cohomology theory. Even better, Brown's representability theorem states that every generalized (co)homology theory comes from a spectrum, so we have a 1:1 correspondence.

The goal of this article is now to define Thom spectra and to give a geometric interpretation of the corresponding homology and cohomology theories, essentially by carrying out the Pontryagin-Thom construction relatively.


Some Preliminaries on Transversality

To understand this article it may help to have seen the proof that framed cobordism \Omega_n^{fr} is isomorphic to stable homotopy groups of spheres, via the Pontryagin-Thom construction, but it is not strictly necessary.

I will assume some technical stuff on transversality, the most important being the
Theorem: Let f : M \to N be a smooth map and Y \subset N a smooth codimension k submanifold, such that f intersects Y transversally (i.e. f maps the tangent bundle of M to a subbundle of the tangent space of N that spans, together with the tangent bundle of Y, the whole tangent bundle of N), then the preimage f^{-1}(Y) is a smooth codimension k submanifold of M.

This theorem follows from the implicit function theorem much like the regular value theorem (by constructing appropriate coordinate charts), and generalizes it (take Y to be a point). It also generalizes the well-known constant rank theorem. To be transversal is a precise way of being "in general position".

The technical heart (in my opinion) of the Pontryagin-Thom construction (over a point) is the
Thom Transversality Theorem: Let f : M \to N be a smooth map and Y \subset N a smooth submanifold, then there exists an arbitraily small perturbation of f (i.e. for any \epsilon > 0 a homotopic map g such that the values are only varying in an \epsilon-ball around each point) which is transversal to Y.

The transversality theorem roughly tells us, that being "in general position" is a generic property, which means that the exceptions are ... well, exceptional. This generalizes the theorems of Brown and Sard that tell us that regular values are dense, in the precise way that the transversal maps are a dense subset of the mapping space.

Spectra and (Co)homology theories

I'm assuming here that you already know the loop space functor. It assigns to a space X its space of (based) loops, topologized as subspace of the path space with the standard compact-open topology.

An \Omega-spectrum E is a sequence of spaces E_n (indexed by natural numbers) with weak homotopy equivalences E_n \to \Omega E_{n+1}. Such objects generalize infinite loop spaces, since E_0 is an infinite loop space, and the extra E_n contain additional information (the difference is precisely the question whether the spectrum is connective, but we won't need that in this article).

To each \Omega-spectrum E one can associate a sequence of contravariant functors E^n : Top \to Sets by E^n(X) := [X,E_n], the homotopy classes of maps from X into the n-th space of the spectrum. One can also associate a sequence of covariant functors E_n : Top \to Sets by E_n(X) := \pi_n(X \wedge E), where X \wedge E is a spectrum with entries (X\wedge E)_n := X \wedge E_n and the homotopy groups are defined as the homotopy groups of the 0-th space of the spectrum for n non-negative, and there is a definition for negative n that shouldn't bother us right now (for the connective spectra aka infinite loop spaces, the negative homotopy groups vanish anyway).

Now one can formally check that the covariant functors form a homology theory, while the contravariant functors form a cohomology theory (both in the sense of Eilenberg-Steenrod axioms), the only nontrivial thing to check is given by the fiber sequence resp. the cofiber sequence.

This term (summer 2012) I gave an expository talk on a theorem in the subject of stable homotopy theory:
Brown Representability Theorem: Every generalized Eilenberg-Steenrod cohomology theory is representable by a spectrum.

I have talk notes on infinite loop spaces, that cover the proof and the preliminary notions mentioned in this section more thoroughly (focusing on the cohomology side).

Classifying spaces

In what follows, we need to know what the classifying space BO(k) of the orthogonal group O(k) is. By definition, if there exists a contractible space E with a free G-action (G some topological group) then the quotient E/G is called classifying space of G, also denoted by BG.

For finite groups, this coincides with Eilenberg-Mac Lane spaces, but there is a considerable conceptual difference, which becomes visible for topological groups.
One has to prove that such a thing actually exists, and there are various constructions, notably the Bar construction. Instead of working in full generality, I just want to use a concrete model:

BO(k) := Gr(\infty,k), the infinite Grassmannian of k-subspaces in some larger space. It is obtained as an inductive limit over the inclusions Gr(n,k) \to Gr(m,k) for m \geq n, where Gr(n,k) is the space of all k-dimensional sub-vector spaces in \mathbb{R}^n.

There are inclusions BO(k) \to BO(k+1) coming from inclusions Gr(n,k) \to Gr(n+1,k+1)
that are (in a certain sense) corresponding to inclusions O(k) \to O(k+1) (both non-canonical, but easily fixed once and for all).

The contractible space with O(k)-action is given by the total space of the so-called tautological bundle, which is a vector bundle over Gr(n,k) that has as fiber over a point exactly the subspace of \mathbb{R}^n this point represents. This gives in the limit a vector bundle over Gr(\infty,k), with an obvious O(k)-action.

The terminology "classifying" comes from the fact that homotopy classes from a manifold M into a classifying space BG for some topological group G classify exactly the G-principal bundles up to isomorphism. In particular, using the fact that the isomorphism classes of O(k)-principal bundles are in bijection with all vector bundles, we have

 [M,BO(k)] \simeq \{\text{rank } k \text{ vector bundles on } M\}

and the isomorphism is given by pulling back the tautological bundle along a map M \to BO(k). That's why the tautological bundle is sometimes called universal bundle.

So it makes sense to take a codimension k submanifold M \subset \mathbb{R}^N, look at its normal bundle \nu over \mathbb{R}^N (which is of rank k) and assign to it a classifying map \tilde{\nu} : \mathbb{R}^N \to BO(k) (actually only a homotopy class, but we can always choose representatives).


We define X-structures, which allow an easy setup to define general Thom spectra later, out of the construction for BO (i.e. real vector bundles). If the X-business is too much for you, stick to X=BO. The following I learned from Switzer's book.

Definition: Let X be a sequence of spaces X_n together with maps X_n \to X_{n+1} and fibrations X_n \to BO(n) that commute with the canonical map BO(n) \to BO(n+1). An X-structure on a smooth manifold M is a pair (h,\tilde{\nu}) such that h : M \to \mathbb{R}^{n+k} is an embedding with normal bundle classified by \nu : M \to BO(k) and \tilde{\nu} : M \to X_k is a lifting of \nu along the fibration X_k \to BO(k).
An X-structure induces for all m\geq k maps h_m = i\circ h : M \to \mathbb{R}^{n+m} and \tilde{\nu}_m = B i_{m-1} \circ \cdots \circ B i_k \circ \nu : M \to BO(m).
Two X-structures (h,\tilde{\nu} : M \to X_k) and (h',\tilde{\nu}' : M \to X_{k'}) are called equivalent, if there is some k'' \geq max(k,k') and a translation T : \mathbb{R}^{n+k''} \to \mathbb{R}^{n+k''} such that h'_{k''} = T \circ h_{k''} and \tilde{\nu}'_{k''} is homotopic to \tilde{\nu}_{k''} through liftings (i.e. H : M \times I \to X_{k''} commutes with the fibration X_{k''} \to BO(k'') for all times t).
An X-manifold is a smooth manifold M together with an equivalence class of X-structures.

The empty set will be regarded as n-manifold for all n, with unique X-structure.

If this X confuses you, you can take as concrete examples for X \to BO the cases \{\ast\} \to BO (which yields framed (co)bordism, as studied by Pontryagin) and id : BO \to BO (which yields ordinary (co)bordism).

Definition: A map of X-manifolds is a smooth map f : M \to M' between manifolds with X-structures (h,\tilde{\nu}) and (h',\tilde{\nu}') such that there is a translation T with h' \circ f = T \circ h and there exists a homotopy \tilde{\nu}' \circ f \simeq \tilde{\nu} that lifts \nu' \circ f = \nu.

Definition: Let M_1,M_2 be two closed n-dimensional X-manifolds. They are called X-cobordant, M_1 \sim_X M_2, if there exist (n+1)-dimensional compact X-manifolds W_1,W_2 such that M_1 \sqcup \partial W_2 \simeq \partial W_1 \sqcup M_2 are X-diffeomorphic (with the induced X-structures on the boundaries).

This is easily seen to be an equivalence relation, we write \Omega_n^X or \Omega_n^X(pt) for the classes. One can also show that disjoint union gives \Omega_n^X an abelian group structure with \emptyset as neutral element.

Thom spectra (for X-structures)

Now we're going to construct the objects I want to investigate. For a general first idea what Thom spaces are about, you can have a look at my previous post on Thom spaces and their interpretation as twisted suspensions.

Definition: Let \xi : E(\xi) \to B(\xi) be a rank n vector bundle. Taking any inner product on the fibers, we can consider \xi an O(n)-bundle and thus define the disk bundle D(\xi) := \{v \in E(\xi) | |v| \leq 1\} and the sphere bundle S(\xi) := \{v \in E(\xi) | |v| = 1\}. Taking the quotient of the total spaces yields the Thom space

 M(\xi) := D(\xi)/S(\xi)

which comes with a natural projection D(\xi) \to M(\xi).

If the base of a bundle has a CW structure, so has the Thom space (and one can describe the structure precisely).

The Thom construction extends to maps, since any map of O(n)-bundles f : \xi \to \eta satisfies f(D(\xi)) \subset D(\eta) and f(S(\xi))\subset S(\eta), so we have

 M(f) : M(\xi) \to M(\eta).

Proposition: For vector bundles \xi over Y and \eta over Z, there is a natural homeomorphism

 M(\xi) \wedge M(\eta) \xrightarrow{\sim} M(\xi \times \eta).

This is essentially the homeomorphism

 \dfrac{D(\xi) \times D(\eta)}{D(\xi)\times S(\eta) \cup S(\xi) \times D(\eta)} \simeq \dfrac{D(\xi\times\eta)}{S(\xi\times\eta)}.

As a corollary, look at \xi \oplus \epsilon^n as \xi \times \epsilon^n with \epsilon^n a trivial bundle (first regarded over the same space as \xi but then as bundle over a point), then we have

 M(\xi \oplus \epsilon^n) = M(\xi) \wedge M(\epsilon^n) = M(\xi) \wedge S^n = \Sigma^n M(\xi).

Definition: Let X = \{X_n,g_n,f_n\} be an X-structure and denote by \gamma_n the universal (tautological) O(n)-bundle over BO(n). Pulling it back to X we have \omega_n := f_n^\ast \gamma_n, which satisfies

 g_n^\ast \omega_{n+1} = g_n^\ast f_{n+1}^\ast \gamma_{n+1} \simeq f_{n}^\ast (Bi_n)^\ast \gamma_{n+1} \simeq f_n^\ast (\gamma_n \oplus \epsilon^1) \simeq \omega_n \oplus \epsilon^1,

so g induces a bundle map \omega_n \oplus \epsilon^1 \to \omega_{n+1} and on Thom spaces
M(g_n) : \Sigma M(\omega_n) \to M(\omega_{n+1}).
This is the data for a spectrum MX and it is customary to use the notation MX_n := M(\omega_n) for the Thom spectrum. To get an honest \Omega-spectrum (to calculate homotopy groups), one still needs to stabilize, i.e. take MX_n := \Sigma^\infty \Omega^\infty MX_n. We will not do this, but rather represent a homotopy class in \pi_k(MX) by a map S^{k+N}(MX_N) for some very large N, which amounts to the same.

Let's see what we've got so far: we have defined various spectra associated to X-structures. We also have a notion of being X-cobordant. The following will bring these threads together.

Thom's theorem and (co)bordism (co)homology

Thom's theorem over a point

Theorem:  \Omega_\ast^X(pt) \simeq \pi_\ast(MX).

We first describe a map \Phi defined on the X-diffeomorphism classes of X-manifolds of dimension n into \pi_n(MX) = \pi_{n+k}(MX_k), then we show that it factors through a homomorphism  \Omega_\ast^X \to \pi_\ast(MX). This map is shown to be surjective and with similar arguments, that it is also injective.

Given a closed smooth n-dimensional manifold M with X-structure (h,\tilde{\nu}), where h : M \to \mathbb{R}^{n+k}, we regard S^{n+k} as the 1-point compactification of \mathbb{R}^{n+k} and the normal disk bundle D(\nu) of M in \mathbb{R}^{n+k} as a tubular neighbourhood of M in S^{n+k} \setminus \{s_0\}. We define a map g : S^{n+k} \to M(\nu) (which represents a homotopy class of the Thom space of \nu) by letting it be the projection \pi : D(\nu) \to M(\nu) on the subset D(\nu) \subset S^{n+k} \setminus \{s_0\} and the constant map to the basepoint on the complement. This is continuous since the boundary of D(\nu) is also sent to the basepoint by construction. By composing g with M(\tilde{\nu}) : M(\nu) \to MX_k we get a map M(\tilde{\nu})\circ g =: f_M^k : (S^{n+k},s_0) \to (MX_k,\ast), thus a map of spectra f_M : S^n \to MX and define \Phi(M):=[f_M] \in \pi_n(MX).

Now we show that the disjoint union of two n-dimensional X-manifolds (M,h,\tilde{\nu}), (M',h',\tilde{\nu}') is mapped by \Phi to the sum \Phi(M)+\Phi(M') \in \pi_n(MX).
We may assume h'(M')\cap h(M) = \emptyset in \mathbb{R}^{n+k} by translating the map h away from the image of h' (by virtue of the definition of an X-structure, this still gives the same X-structure). We can even translate h and h' such that one lands entirely in the upper half space and the other in the opposite half space, so that we observe that f_{M\sqcup M'} : S^n \to MX is f_M on the upper hemisphere and f_{M'} on the lower hemisphere. The map f_{M\sqcup M'} thus factors through S^{n} \vee S^{n}, by pinching the equator of S^n to a point.

The next step is to show that \Phi is invariant under X-cobordism. Let (W,h,\tilde{\nu}) be an X-manifold with boundary, where we regard h : W \to \mathbb{R}^{n+k} after translation as embedding into \mathbb{R}^{n+k+1}_+ and thus as embedding into S^{n+k} \times [0,1), with \partial W landing in S^{n+k} \times \{0\}. Again we proceed to obtain a map f_W^k : S^{n+k} \times I \to MX_k that yields a map f_W : S^n \times I \to MX which is a homotopy from f_{\partial W} = f_W(\cdot,0) to \ast = f_W(\cdot,1), so we observe [f_{\partial W}] = 0 \in \pi_n(MX).
In particular, two X-manifolds that are X-cobordant M \sim_X M' via some X-manifold W with boundary \partial W = M \sqcup -M' yield [f_M] - [f_{M'}] = [f_{\partial W}], so we have [f_M] = [f_{M'}] and thus \Phi factors through a homomorphism  \Omega_\ast^X \to \pi_\ast(MX).

For surjectivity of \Phi we take a map f : (S^{n+k},s_0) \to (MX_k,\ast) representing a class in \pi_n(MX) and construct an X-manifold M as codimension k submanifold of S^{n+k} such that f_M \simeq f, i.e. \Phi(M) = [f] \in \pi_n(MX).
To do that, we slightly deform Mf_k \circ f : S^{n+k} \to MO_k such that it is transversal to BO(k), which allows to take M := f^{-1} ( (Mf_k)^{-1} (BO(k)) ) \subseteq S^{n+k}. The homotopy can be lifted to a homotopy of f, since f_k was required to be a fibration. Taking a tubular neighbourhood T of BO(k) inside E(\omega_k) we can carry out the same argument, f taken to be transversal to T and so we get f^{-1}(T) as a tubular neighbourhood of M. This gives us an X-structure on M and at the same time we can see that the map f_M assigned by \Phi to M is homotopic to f.

Injectivity uses the same transversality trick that we just saw. Take two manifolds M,M' \in \Omega_n^X with [f_M] = [f_{M'}] \in \pi_n(MX), so we have a homotopy H : S^{n+k} \to MX_k with H_0 = f_M and H_1 = f_{M'}. With the transversality trick we deform H such that W := H^{-1}(BO(k)) is a submanifold. It is necessarily a dimension n+1 submanifold, since each H_t^{-1}(BO(k)) is a codimension k submanifold of S^{n+k} \times \{t\}. We see that \partial W = M \sqcup -M' and with the tubular neighbourhood trick we get an X-structure on W as well.

Singular manifolds, relative Thom's theorem

Now that we understood the situation over a point, the general case will not be much harder. I will briefly state what we do now:
To any spectrum MX one can not only associate it's homotopy groups \pi_n(MX) but also a (reduced) homology functor Y \mapsto \pi_n(MX \wedge Y). We will write MX_k(Y):=\pi_n(MX \wedge Y) and call it the k-th X-bordism of Y. The question is: what is the (geometric) meaning of the k-th X-bordism of some manifold?

The answer is, that the k-th X-bordism of Y classifies the singular X-manifolds over Y, up to cobordism. The case of Y=pt was solved in the previous subsection, where "singular X-manifold over a point" reduces to "X-manifold".

Definition: A continuous map f : M \to Y from a closed X-manifold M to Y is called singular X-manifold in Y. Two singular X-manifolds f : M \to Y, f' : M' \to Y are X-cobordant if there is a compact X-manifold W with boundary \partial W \simeq M \sqcup -M' together with a continuous map g : W \to Y that restricts to the singular X-manifolds g|M = f, g|M' = f'.

Theorem: \Omega_\ast^X(Y) \simeq MX_\ast(Y).


The strategy is the same as in the previous proof. First I summarize, then we can go through the details:
a) To each compact smooth n-fold (with an X-structure) M with continous map s : M \to Y we assign a map f_s : S^{n+k} \to MO_k \times Y by the Thom space construction (here, one does something different than in the case Y=pt).
b) We compose such a map f_s with the projection MO_k \times Y \to MO_k \wedge Y and also with Mf_k \wedge Y (order doesn't matter), and take homotopy classes. We obtain a map that factors through X-diffeomorphisms

 \Phi : \{\text{singular X-manifolds } s : M \to Y\}/\text{diffeo} \to \pi_{n}(MX \wedge Y).

c) Show that disjoint union of manifolds corresponds to addition in the homotopy group, by the pinching trick (putting one manifold in the upper and the other in the lower hemisphere).
d) \Phi factors through a group homomorphism \Omega_n^X(Y) \to MX_n(Y), since an X-cobordism W of singular manifolds s : M \to Y and s':M'\to Y yields a homotopy \Phi(W) between f_s and f_{s'}.
e) Surjectivity of \Phi is done with the transversality trick: We get a preimage of some [f] by taking a representative f that is transversal to BO_k \wedge Y, and then M := f^{-1}(BO_k\wedge Y) is a manifold with continuous map s := proj_Y \circ f : M \to Y such that f_s = f.
f) Injectivity also uses the transversality trick: For two singular X-manifolds s,s' that get mapped to the same homotopy class, we have a homotopy H between f_s and f_{s'} that comes from a cobordism (essentially by surjectivity of some kind of \Phi).

The difficulties lie in step a) and that one has to keep track of the "singular" thing, i.e. we don't have just manifolds on the left hand side, but continuous maps.

So I explain step a) in more detail now:
Let s : M \to Y be a singular X-manifold. Consider the (n+k)-sphere as one-point compactification S^{n+k} = \mathbb{R}^{n+k} \cup \{\infty\} and define f : S^{n+k} \to MX_k \times Y by f|D(\nu) := M(\tilde{\nu}) \circ ( \pi : D(\nu) \to M(\nu) ) \times (s\circ \nu), where M(\tilde{\nu}) : M(\nu) \to MX_k is the map induced by the X-structure and s \circ \nu is the composition D(\nu) \to M \to Y that assigns to each vector in the normal bundle the image of its footpoint under s. On the complement, we send everything to the basepoint, f|(S^{n+k} \setminus D(\nu)) := \ast. We compose the result with the contraction MX_k \times Y \to MX_k \wedge Y. That's the map f_s. The assignment s \mapsto f_s is well-defined on the level of X-diffeomorphism classes of singular X-manifolds, and we call this map \Phi.

Change of coefficients: Bockstein

Every complex manifold has a complex normal bundle, so it comes with a BU-structure (X is now BU \to BO). This means that we can look at \Omega^{BU}_\ast(Y) \to \Omega^{BO}_\ast(Y) by forgetting this extra structure. At the same time we can look at BU \to BO as inducing a map of spectra MU \to MO that induces homology morphisms MU_n(Y) \to MO_n(Y), that coincide with the map described before.

One can now ask whether two non-complex-cobordant manifolds become real-cobordant, i.e. whether their images under the Bockstein morphism just sketched coincide. One can also ask whether a given real manifold is in the image of the Bockstein morphism.

The new thing is now, that we can use fiber sequence technology to get more information. Since X \to BO is required to be a fibration, we can call the fiber F and get a long exact sequence

 \cdots \to MF_n(Y) \to MX_n(Y) \to MO_n(Y) \to MF_{n-1}(Y) \to \cdots

The connecting morphism in this long exact sequence is sometimes the only one called "Bockstein".

Cobordism Cohomology

I wanted to discuss this in more detail, but then I got exhausted from writing up, so here is a rough sketch:

Cobordism Cohomology can be defined as MX^n(Y) := [S^k \wedge Y, MX_{n+k}] for k large enough. One can try to do the same as for homology, to identify the "geometric" object MX^n(Y) should be isomorphic to: Given a homotopy class [f] \in [S^k \wedge  Y, MX_{n+k}], we can choose a representative f that extends to S^k \times Y \to MX_{n+k} such that it's transversal to BO(n+k) in MX_{n+k} and then f^{-1}(BO(n+k)) is a smooth submanifold of S^k \times Y which becomes a singular X-manifold in Y by projecting to Y. Working out the dimensions, we get \Omega^X_{\dim Y - n}(Y) \simeq MX^{n}(Y).

For a better overview in the special case X=SO you can look at Atiyah: Bordism and Cobordism.


There are various things one can do from this point on.

  • Do the same stuff algebraically, as in Morel-Levine's book on algebraic cobordism.
  • Look at framed cobordism to get some knowledge about stable homotopy groups of spheres (Pontryagin's observation)
  • Look at complex cobordism and the Adams-Novikov spectral sequence to get even more knowledge of stable homotopy groups. This is currently discussed in a rather long series of blog posts by Akhil Mathew.
  • Use a better understanding of cobordisms to get some knowledge about mapping class groups, as in Madsen-Weiss.
  • Forget all this stuff (maybe you didn't read it carefully in the first place, so why bother?)

Category: English, Mathematics

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2 Responses

  1. Dear Dr. Voelkel
    I learned many things from your pretty note on "Bordism
    and Cobordism". I just want to know the precise title
    of th "Switzer's book" that you have referenced in the text.
    Sincerely yours,
    Sahand Raman

  2. (... not a Dr. yet ...)

    Switzer's book is called "Algebraic Topology: Homotopy and Homology".