Recap: bordism group

An oriented singular $n$-manifold in $X$is a map $f:M^n\to X$ where $M$ is a finite disjoint union of $n$-dimensional smooth manifolds. The empty set is an admissible oriented singular $n$-manifold.

Two oriented singular manifolds $f_i:M_i^n\to X$ $i=1,2$ are bordant in $X$ if exists a map $F:W^{n+1}\to X$ ($W^{n+1}$ smooth oriented) such that $M_1\sqcup -M_2\subset \partial W$ (orientation is induced), $F|_{M_1\sqcup -M_2} \equiv f_1\sqcup f_2$.

This defines an equivalence relation and the equivalence classes $[M^n,f]$ of oriented singular $n$-manifolds up to bordism define the oriented $n$-bordism group of $X$ denoted $\Omega_n(X)$. The (abelian) group structure is given by the operation $[M_1,f_1]+ [M_2,f_2] = [M_1\sqcup M_2, f_1\sqcup f_2]$. If we forget all the orientation requirement above, we get the (unoriented) n-bordism group of $X$ denoted as $\eta_n(X)$. With $\Omega_n$ is intended $\Omega_n(pt)$ i.e. $X={*}.$

Question 1:Thom's theorem

Thom discovered an isomorphism of groups: $$\phi:\Omega_n \to \pi_{n+k}(MSO_k)$$ where $k>n+1$ and $MSO_k$ is the Thom space of the universal $SO_k$ bundle $ESO_k\to BSO_k$. Roughly $\phi([M])$ is constructed from an embedding $M\to \mathbb{R}^{n+k}$, taking the classifying map for the normal bundle $\tau_\nu:M\to BSO_k$ and extending it to a map $\mathbb{R}^{n+k}\to ESO_k$ that is constant in a neighbourhood of $\infty$. This yelds a map $\mathbb{S}^{n+k}\to MSO_k$.

Question 1: This works for $\Omega_n(pt)$, what kind of isomorphism do we have for $\Omega_n(X)$? How is it defined? References? I hope there is a geometric definition similar to the one I sketched above (that clarifies the relation with the normal bundle of an embedding for example).

Question 2: Thom-Pontryagin theorem on characteristic numbers

We have the following theorem: $[M]= 0 \in \Omega_n$, i.e. $M$ bounds an oriented manifold iff all the Stiefe-Whitney numbers and all the Pontryagin numbers of $M$ vanish: $$\langle w_{i_1}(M)\cup \cdots \cup w_{i_{k'}}(M), [M]\rangle = 0\ \ \ \ \ \langle p_{i_1}(M)\cup \cdots \cup p_{i_k}(M), [M]\rangle = 0 $$

For $\eta_n(X)$ things are similar: $[M,f]= 0\in \eta_n(X)$ iff $$\langle w_{i_1}(M)\cup \cdots \cup w_{i_k}(M)\cup f^*(h), [M]\rangle = 0 \in \mathbb{Z}_2\ \ \ \forall h \in H^*(X,\mathbb{Z}_2) \text{ and SW classes}$$ i.e. all SW numbers of the map $f$ vanish.

Curiously, for $\Omega_n(X)$ the only statement I've found requires an additional assumption: if all torsion classes in $H_*(X,\mathbb{Z})$ have order exactly $2$, then $[M,f]= 0 \in \Omega_n(X)$ iff the SW and Pontryagin numbers of $f$ vanish. Also usually it's required $X$ to be (h-equivalent to) a finite CW complex.

Question 2a: how to prove the statement for $\eta(X)$ and $\Omega(X)$? What are some good references? Is there any geometric interpretation of why the vanishing of the characteristic numbers entails the extendability of the map $f$?

Question 2b: Can we get rid of the assumption on the torsion of $H_*(X)?$ Why is it needed?

Question 3: What important achievements regarding bordism theory has been reached after the 60s?

Note: the reference I have used here is Conner P.E and E.E. Floyd: Differentiable Periodic Maps. Bull. Am. Math. Soc. 68, 76-86 (1962).

  • 2
    $\begingroup$ Question 1 and the second part of 2b have more or less trivial answer. A1. Oriented bordism is a generalized homology represented by the spectrum MSO. You can find this in any decent textbook of cobordism. For example Rudyak. A2b') $H_(*,\mathbf{Z})$ has no torsion. I have a feeling that the first part of the question 2b should have a trivial answer, if we are looking for an example of the pair $([M,f],X)$ and not an example of $[M,f]$ for fixed $X$. $\endgroup$ – user43326 Mar 7 at 15:00

For question 2b, the answer is that elements of $H^n(X; A)$ determine bordism invariants $\Omega_n^{\mathrm{SO}}(X)\to A$, and if $H^*(X)$ contains $p$-torsion for $p$ odd, these can't be interpreted as Stiefel-Whitney or Pontrjagin numbers. A simple example is $\Omega_1^{\mathrm{SO}}(B\mathbb Z/3)$, the bordism group of oriented 1-manifolds with a principal $\mathbb Z/3$-bundle. A circle with a nontrivial principal $\mathbb Z/3$-bundle does not bound, but all Stiefel-Whitney and Pontrjagin numbers here vanish. In fact pushing the fundamental class forward defines an isomorphism $\Omega_1^{\mathrm{SO}}(B\mathbb Z/3)\to H_1(B\mathbb Z/3)\cong\mathbb Z/3$.

Question 2a is harder: the proofs are generally not geometric, but rather homotopical. One takes the spectrum $\mathit{MSO}$ (built from the spaces $\mathit{MSO}_k$) and splits it as a wedge sum of other spectra we understand. Such a splitting entails a lot of messy calculations with the Steenrod algebra and as far as I know there isn't a geometric proof. For unoriented bordism, Thom split $\mathit{MO}$ splits as a wedge sum of shifts of Eilenberg-Mac Lane spectra. The isomorphism is built out of the cohomology of $\mathit{BO}$, which ultimately means it can be interpreted as sending a manifold to its Stiefel-Whitney numbers. I don't know what the best reference is, but Thom's thesis is the original reference.

For oriented bordism, the analogous splitting result holds only after localizing. Thom showed that if you rationalize, you can split $\mathit{MSO}$ into a sum of Eilenberg-Mac Lane spectra, and the map sends a manifold to its Pontrjagin numbers. If you complete at 2, the calculation is harder; there is again a splitting, interpretable as sending a manifold to its Pontrjagin and Stiefel-Whitney numbers, and I think this was shown by Wall, again homotopically rather than geometrically. (At odd primes, one uses a generalized cohomology theory called Brown-Peterson cohomology to split $\mathit{MSO}$, and again the proof is homotopical.) I don't know of a good reference digging into these calculations other than the original papers, but Manifold Atlas has a good summary and list of references.

For question 3, some of the most exciting achievements have been the study of bordism categories, such as the work of Galatius-Madsen-Tillmann-Weiss on the homotopy types of bordism categories; the cobordism hypothesis; and some of the applications to topological field theory, homological stability, and so on. These results use and strengthen Pontrjagin-Thom theory.

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