Topological invariance of Stiefel-Whitney classes for open smooth manifolds

Question

It is well known that Stiefel-Whitney classes are homotopy invariant for closed smooth manifolds. But in the case of open manifolds even $w_1$ is not a homotopy invariant (take just open cylinder and open Mobius strip). Therefore, the following question naturally arises.

Conjecture 1. Suppose that $f\colon M^n \to N^n$ is a purely continuous homeomorphism of two connected open manifolds homotopy equivalent to a finite polyhedron. Then $f^*w_k(N^n) = w_k(M^n)$ for all $1\le k\le n.$

The conjecture is trivially true for $w_1$ (a loop preserve or change the orientation), and for $w_n=0$.

Due to the fact that the direct product with euclidian space preserve S-W classes, we can assume that the dimension $n$ is as big as we want.

Also, the prominent theorem of Thom states that any mod 2 homology class of an arbitrary connected topological space can be realized by the image of the fundamental class of some connected closed smooth (possibly unorientable) manifold. Fix any $k\ge 2$. Suppose $g\colon L^k \to M^n$ is a continuous mapping ($L^k$ is connected, closed and smooth). Then $w_k(g_*[L^k])$ depends only on the behavior of the manifold $M^n$ around the image $g(L^k)\subset M^n$.

Therefore, it is easy to see that the above conjecture is equivalent to the following

Conjecture 2. Fix any $k\ge 2.$ Then for some $N_0\gg 1$ and any $n\ge N_0$ the following statement holds true. Suppose $\overline{M}^n$ is a smooth connected $n$-manifold with a connected boundary. Suppose also, that $N^n$ is a smooth open connected $n$-manifold, which is homeomorphic to the interior $M^n := \mathrm{int}\overline{M}^n$. Then for any purely continuous homeomorphism $f\colon M^n \to N^n$ one has $f^*w_k(N^n) = w_k(M^n)$.

I can prove the Conjecture 2 for the case when the Whitehead torsion $\mathrm{Wh}(\pi_1 \partial \overline{M}^n)$ is zero.

Answer 1

I think that your conjecture 1 is correct. The point is that there is a topological version of the tangent bundle, developed by Milnor, called a microbundle. It has the property (kind of obvious from the definition) that it pulls back under homeomorphisms. Milnor's original paper on microbundles points out that you can define Stiefel-Whitney classes for microbundles `using the Thom definition'. This means using the Steenrod squares applied to the Thom class, as explained in Milnor-Stasheff. (In particular, the SW classes of the microbundle defined in this way are equal to the SW classes of the usual tangent bundle.)

Milnor doesn't give details; this is just a quick remark in his paper. So there are some things you'd want to check, mainly naturality of the SW classes of the tangent microbundle under homeomorphisms. Probably this argument is documented somewhere.

Answer 2

The topological invariance of Stiefel-Whitney classes of smoothable manifolds is just a classic theorem of Thom, written in one of his most famous papers. It was pointed out to me by Alexander A. Gaifullin, and I am very grateful to him.

Rene Thom, Espaces fibres en spheres et carres de Steenrod, Annales Scientifiques de l'E.N.S. 3e serie, tome 69 (1952), p. 109-182.

Theorem III.8. Suppose $M^n$ is a smoothable connected (paracompact) $n$-manifold (it could even be of an infinite homology type). Suppose also that $\Sigma$ is a smooth structure on $M^n$. Then Stiefel-Whitney classes of a tangent bundle of $M^n$ does not depend on $\Sigma$.

This theorem is written on the page 151. The proof is just as in my previous answer. Also, on the pages 151-152 of the paper Thom gives the Wu formula for S-W classes of a closed manifold (this proves the homotopy invariance of S-W classes in this case).

Answer 3

Here is the proof of the Conjecture 1. The idea of the proof came from the upper answer by Danny Ruberman and by private communication with Semen Podkorytov. I am very grateful to them!

To prove this Conjecture we need some auxiliary observations. Let $M^{n+k}$ be a connected TOP $(n+k)$-manifold and $N^n\subset M^{n+k}$ be a closed connected TOP submanifold of dimension $n$. Both manifolds $N^n$ and $M^{n+k}$ are assumed to be of a finite homotopy type, but they may be nonorientable, and the inclusion $N^n\hookrightarrow M^{n+k}$ may be wild (non locally flat).

The following definition-proposition can be found in the classical textbook by A.Dold (Lectures on algebraic topology, 2ed., Chapter VIII, Corollary 11.20). Let the coefficient ring be $\mathbb{Z}_2:=\mathbb{Z}/2\mathbb{Z}$.

Definition-Proposition. The cohomology group $H^i(M,M-N)$ is zero for $0\le i \le k-1$. The group $H^k(M,M-N)$ consists only one nonzero element $u_{M,N}$, which is called the Thom class of the pair $(M,N)$. Suppose $r\colon M' \to N$ is a retraction of an open domain $N\subset M' \subset M$. Then the composition $$ H^q(N) \xrightarrow{r^*} H^q(M') \xrightarrow{\smile u_{M',N}} H^{q+k} (M',M' - N) \stackrel{\mathrm{exc}}{\cong} H^{q+k} (M,M-N) $$ is an isomorphism for all $0\le q \le n$. Moreover, this isomorphism is independent of the choice of the domain $M'$ and a retraction $r\colon M' \to N$. This mapping is called the Thom isomorphism.

Due to this fact we can give the following definition of Stiefel-Whitney classes of a connected topological manifold $M^n$ (closed or open and of a finite homotopy type).

Definition 1. Let us consider the pair $M^n \stackrel{\Delta}{\hookrightarrow} M^n\times M^n$. By $\phi_{M}$ we will denote the Thom isomorphism $\phi_{M}\colon H^*(M) \to H^{*+n} (M\times M, M\times M - \Delta(M))$. We define the Stiefel-Whitney classes $w_k(M)$ by the formula $w_k(M) := \phi_{M}^{-1} \mathrm{Sq}^k \phi_{M} (1), 1\le k \le n$. Here, $\phi_{M}(1)$ is equal to the Thom class $u_{M\times M, \Delta(M)}$.

Now, suppose that a connected topological manifold $M^n$ (closed or open and of a finite homotopy type) is smoothable and $\Sigma$ is a smooth structure on $M^n$. Then $\Sigma\times \Sigma$ is a corresponding smooth structure on $M^n\times M^n$, and $\Delta(M^n)\subset M^n\times M^n$ is a smooth submanifold. Moreover, the normal vector bundle $\xi_{M}$ of the submanifold $\Delta(M^n)\subset M^n\times M^n$ is isomorphic to the tangent bundle $\tau_{M}$ of $M^n$.

Let us fix some Riemannian metric $g_{M\times M}$ on the manifold $M^n\times M^n$. Then we can take some small tubular neighborhood $E_{M}$ of $\Delta(M^n)\subset M^n\times M^n$ and the corresponding retraction $r_{M}\colon E_{M} \to \Delta(M)$. The retraction $r_{M}$ is a strong deformation retraction. Therefore, $r_{M}^*\colon H^*(\Delta(M))\to H^*(E_{M})$ is an isomorphism.

Now, it is evident, that we are in the standard situation of vector bundles and their Stiefel-Whitney characteristic classes. We have the standard Thom class $u_{M}\in H^n(E_{M}, E_{M} - \Delta(M))$ of the vector bundle $\xi_{M}\cong \tau_{M}$, and the standard Thom isomorphism $\phi_{M}\colon H^*(M) \to H^{*+n} (E_{M}, E_{M} - \Delta(M))$, which is defined by the composition $$ H^q(M) = H^q(\Delta(M)) \xrightarrow{r_{M}^*} H^q(E_{M}) \xrightarrow{\smile u_{M}} H^{q+n} (E_{M}, E_{M} - \Delta(M)), \ 0\le q\le n. $$

Finally, we get the Stiefel-Whitney classes $w_k(M)$ of the tangent bundle $\tau_{M}$ by the Thom formula $w_k(M) := \phi_{M}^{-1} \mathrm{Sq}^k \phi_{M} (1), 1\le k \le n$. But, due to the Definition-Proposition and the Definition 1, it is clear, that the Thom class, the Thom isomorphism and the Stiefel-Whitney classes obtained from the smooth structure $\Sigma$ on $M^n$ coincide with the topological Thom class, Thom isomorphism and Stiefel-Whitney classes of $M^n$.

Evidently, these observations prove the Conjecture 1.

Answer 4

In fact, conjecture 2 can be strengthened. Recall that the normal spherical fibration of a manifold $M$ is obtained by fiberwise one point compactifying the normal bundle of $M$. Similarly, the tangential spherical fibration is given by fiberwise one point compactifying the tangent bundle.

If $M,N$ are homeomorphic to the interior of a compact manifold (i.e. tame), then any proper homotopy equivalence $f: M \rightarrow N$ stably pulls back the normal spherical fibrations by corollary 3.4 in Wall's Poincare Complexes I. Since the normal spherical fibration is the stable inverse of the tangential spherical fibration, we also conclude that proper homotopy equivalences pull back the stable tangential spherical fibration.

Since the Stiefel-Whitney classes can be defined in terms of the Thom isomorphism, the fact that spherical fibrations also have a Thom isomorphism implies that Stiefel-Whitney classes only depend on the underlying stable spherical fibration. All together this implies that properly homotopy equivalent, tame manifolds have the same Stiefel-Whitney classes.