Is $[N]^\#([N])$ congruent to $w_n(\nu_N)([N])$ mod $2$, where $\nu_N$ is the normal bundle of the embedding of $N$ in $M$?

Question

Let $M$ be a closed, smooth, orientable $2n$-manifold, and let $N$ be a closed, smooth, orientable $n$-submanifold. Let $[N]^\#$ denote the cohomology class (Poincaré) dual to the homology class $[N]$. Geometrically, if $N'$ is another $n$-submanifold, $[N]^\#([N'])$ counts the number of intersections of $N'$ with $N$ (after perturbing them to be in general position), counted with sign. Is $[N]^\#([N])$ congruent to $w_n(\nu_N)([N])$ mod $2$, where $\nu_N$ is the normal bundle of the embedding of $N$ in $M$?

Answer 1

The answer is yes for the following reason: $\omega_n$ is the Euler class mod 2

Now use e.g.

• Theorem 4.7 here which says $e(\nu_N)([N])$ counts number of intersections, or
• you argue that the Thom class of $\nu_N$ in $M$ is the Poincaré dual of $N$ (follows from this exercise). Hence, by pulling back to the cohomology of $N$ the result follows.

You should easily be able to write down the explicit formulas, where you will only need the above facts and naturality. Also note that both of the above arguments are closely related.