Let $M$, $N$ be two manifolds without boundary (it is possible that $M$, $N$ have different dimensions, or one is compact but the other is not, etc.). Suppose $M\simeq N$ ($M$ is homotopy equivalent to $N$). Can we obtain that the Stiefel-Whitney class of $M$ and the Stiefel-Whitney class of $N$ are equal?
My idea: From Wu's formula, the relations of Stiefel-Whitney class are given by Steenrod operations, which are invariant under homotopy equivalence. The following is from A Concise Course in Algebraic Topology, J.P. May, p. 189: