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.