Thom's result and Poincaré duality

Question

I am interested in singularity theory by topology. I want to understand following results.

$f$ is a smooth map of a closed surface $M$ which has only fold points and cusps as its singularities. Suppose that a closed curve $c$ in $M$ intersects a singular set $S(f)$ transversely at a finite number of points.

Then the number of intersection points is odd if and only if $c$ is orientation reversing:

i.e., if and only if $\{w_1(M), [c]\}= 1$, where $w_1(M) \in H^1(M; Z_2)$ is the first Stiefel-Whitney class of $M$, $[c] \in H_1(M;Z_2)$ is the $Z_2$-homology class represented by $c$, and $\{,\}$ is the Kronecker product. $H^1$ is first cohomology and $H_1$ is first homology and $Z_2$ is order $2$ cyclic group.

Above statement is Thom's result which states that the Poincare dual to the $Z_2$-homology class represented by $S(f)$ coincides with $w_1(M)$.

Question

How Thom's result is used for above statement? I want to know in detail. However, I do know little characteristic classes.

Thank you for your considerations.

Answer 1

Let me attempt to answer your question as I understand it.

Let $x\in H^1(M)$ be the Poincaré dual of $[c]\in H_1(M)$ (all (co)homology groups are with $\mathbb{Z}_2$ coefficients). The result of Thom you state is that $w_1(M)\in H^1(M)$ is the Poincaré dual of $[S(f)]\in H_1(M)$.

It is well known that the cup product in cohomology coincides under Poincaré duality with the intersection of transverse representatives of homology classes. It follows that the parity of the number of intersection points of $S(f)$ with $c$ equals the Kronecker product $\langle w_1(M)\cup x,[M]\rangle\in\mathbb{Z}_2$.

Now use the identity $\langle \alpha\cup \beta,\gamma \rangle = \langle \alpha, \beta \cap \gamma\rangle$ relating cup, cap and Kronecker products (see any good Algebraic Topology book, eg Switzer) to conclude that $$\langle w_1(M)\cup x,[M]\rangle = \langle w_1(M),x\cap [M]\rangle = \langle w_1(M),c\rangle$$ as required.

Added in response to OP's comment: One can think of the first Stiefel-Whitney class $$w_1(M)\in H^1(M;\mathbb{Z}_2)\cong\mathrm{Hom}(H_1(M),\mathbb{Z}_2)\cong\mathrm{Hom}(\pi_1(M),\mathbb{Z}_2)$$ as the homomorphism which assigns the value $1$ (resp. $0$) to a loop in $M$ if it is orientation reversing (resp. orientation preserving). I don't know a written reference for this off the top of my head, but I'm sure there are many.