Monday, July 23rd, 2012 | Author: Konrad Voelkel
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 such that is the Thom space of a universal bundle over some classifying space (I will explain that later) and is homotopy equivalent to . Homotopy theorists like to call such a sequence then a
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 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 be a smooth map and a smooth codimension submanifold, such that intersects transversally (i.e. maps the tangent bundle of to a subbundle of the tangent space of that spans, together with the tangent bundle of , the whole tangent bundle of ), then the preimage is a smooth codimension submanifold of .
This theorem follows from the implicit function theorem much like the regular value theorem (by constructing appropriate coordinate charts), and generalizes it (take 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 be a smooth map and a smooth submanifold, then there exists an arbitraily small perturbation of (i.e. for any a homotopic map such that the values are only varying in an -ball around each point) which is transversal to .
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 its space of (based) loops, topologized as subspace of the path space with the standard compact-open topology.
An -spectrum is a sequence of spaces (indexed by natural numbers) with weak homotopy equivalences . Such objects generalize infinite loop spaces, since is an infinite loop space, and the extra 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 -spectrum one can associate a sequence of contravariant functors by , the homotopy classes of maps from into the -th space of the spectrum. One can also associate a sequence of covariant functors by , where is a spectrum with entries and the homotopy groups are defined as the homotopy groups of the -th space of the spectrum for non-negative, and there is a definition for negative 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).
In what follows, we need to know what the classifying space of the orthogonal group is. By definition, if there exists a contractible space with a free -action ( some topological group) then the quotient is called classifying space of , also denoted by .
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:
, the infinite Grassmannian of -subspaces in some larger space. It is obtained as an inductive limit over the inclusions for , where is the space of all -dimensional sub-vector spaces in .
There are inclusions coming from inclusions
that are (in a certain sense) corresponding to inclusions (both non-canonical, but easily fixed once and for all).
The contractible space with -action is given by the total space of the so-called tautological bundle, which is a vector bundle over that has as fiber over a point exactly the subspace of this point represents. This gives in the limit a vector bundle over , with an obvious -action.
The terminology "classifying" comes from the fact that homotopy classes from a manifold into a classifying space for some topological group classify exactly the -principal bundles up to isomorphism. In particular, using the fact that the isomorphism classes of -principal bundles are in bijection with all vector bundles, we have
and the isomorphism is given by pulling back the tautological bundle along a map . That's why the tautological bundle is sometimes called universal bundle.
So it makes sense to take a codimension submanifold , look at its normal bundle over (which is of rank ) and assign to it a classifying map (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 together with maps and fibrations that commute with the canonical map . An
An X-structure induces for all maps and .
Two X-structures and are called
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 the cases (which yields framed (co)bordism, as studied by Pontryagin) and (which yields ordinary (co)bordism).
Definition: Let be two closed n-dimensional X-manifolds. They are called
This is easily seen to be an equivalence relation, we write or for the classes. One can also show that disjoint union gives an abelian group structure with 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 be a rank n vector bundle. Taking any inner product on the fibers, we can consider an -bundle and thus define the disk bundle and the sphere bundle . Taking the quotient of the total spaces yields the
which comes with a natural projection .
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 -bundles satisfies and , so we have
Proposition: For vector bundles over and over , there is a natural homeomorphism
This is essentially the homeomorphism
As a corollary, look at as with a trivial bundle (first regarded over the same space as but then as bundle over a point), then we have
Definition: Let be an X-structure and denote by the universal (tautological) -bundle over . Pulling it back to X we have , which satisfies
so induces a bundle map and on Thom spaces
This is the data for a spectrum and it is customary to use the notation for the
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
We first describe a map defined on the X-diffeomorphism classes of X-manifolds of dimension n into , then we show that it factors through a homomorphism . This map is shown to be surjective and with similar arguments, that it is also injective.
Given a closed smooth n-dimensional manifold with X-structure , where , we regard as the 1-point compactification of and the normal disk bundle of in as a tubular neighbourhood of in . We define a map (which represents a homotopy class of the Thom space of ) by letting it be the projection on the subset and the constant map to the basepoint on the complement. This is continuous since the boundary of is also sent to the basepoint by construction. By composing with we get a map , thus a map of spectra and define .
Now we show that the disjoint union of two n-dimensional X-manifolds is mapped by to the sum .
We may assume in by translating the map away from the image of (by virtue of the definition of an X-structure, this still gives the same X-structure). We can even translate and such that one lands entirely in the upper half space and the other in the opposite half space, so that we observe that is on the upper hemisphere and on the lower hemisphere. The map thus factors through , by pinching the equator of to a point.
The next step is to show that is invariant under X-cobordism. Let be an X-manifold with boundary, where we regard after translation as embedding into and thus as embedding into , with landing in . Again we proceed to obtain a map that yields a map which is a homotopy from to , so we observe .
In particular, two X-manifolds that are X-cobordant via some X-manifold with boundary yield , so we have and thus factors through a homomorphism .
For surjectivity of we take a map representing a class in and construct an X-manifold as codimension submanifold of such that , i.e. .
To do that, we slightly deform such that it is transversal to , which allows to take . The homotopy can be lifted to a homotopy of , since was required to be a fibration. Taking a tubular neighbourhood of inside we can carry out the same argument, taken to be transversal to and so we get as a tubular neighbourhood of . This gives us an X-structure on and at the same time we can see that the map assigned by to is homotopic to .
Injectivity uses the same transversality trick that we just saw. Take two manifolds with , so we have a homotopy with and . With the transversality trick we deform such that is a submanifold. It is necessarily a dimension n+1 submanifold, since each is a codimension k submanifold of . We see that 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 one can not only associate it's homotopy groups but also a (reduced) homology functor . We will write and call it the k-th X-bordism of . 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 classifies the
Definition: A continuous map from a closed X-manifold to is called
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) with continous map we assign a map by the Thom space construction (here, one does something different than in the case ).
b) We compose such a map with the projection and also with (order doesn't matter), and take homotopy classes. We obtain a map that factors through X-diffeomorphisms
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) factors through a group homomorphism , since an X-cobordism of singular manifolds and yields a homotopy between and .
e) Surjectivity of is done with the transversality trick: We get a preimage of some by taking a representative that is transversal to , and then is a manifold with continuous map such that .
f) Injectivity also uses the transversality trick: For two singular X-manifolds that get mapped to the same homotopy class, we have a homotopy between and that comes from a cobordism (essentially by surjectivity of some kind of ).
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 be a singular X-manifold. Consider the (n+k)-sphere as one-point compactification and define by , where is the map induced by the X-structure and is the composition that assigns to each vector in the normal bundle the image of its footpoint under . On the complement, we send everything to the basepoint, . We compose the result with the contraction . That's the map . The assignment is well-defined on the level of X-diffeomorphism classes of singular X-manifolds, and we call this map .
Change of coefficients: Bockstein
Every complex manifold has a complex normal bundle, so it comes with a -structure (X is now ). This means that we can look at by forgetting this extra structure. At the same time we can look at as inducing a map of spectra that induces homology morphisms , 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 is required to be a fibration, we can call the fiber and get a long exact sequence
The connecting morphism in this long exact sequence is sometimes the only one called "Bockstein".
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 for large enough. One can try to do the same as for homology, to identify the "geometric" object should be isomorphic to: Given a homotopy class , we can choose a representative that extends to such that it's transversal to in and then is a smooth submanifold of which becomes a singular X-manifold in by projecting to . Working out the dimensions, we get .
For a better overview in the special case 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?)