TITLE: Topological invariance of Stiefel-Whitney classes for open smooth manifolds
QUESTION [7 upvotes]: 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.

REPLY [4 votes]: 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).