In Milnor's characteristic class, he defines the Stiefel-Whitney class as follows:
Let $M$ be a closed possibly disconnected smoooth $n$-manifold. Using mod $2$ coefficients there is a unique fundamental homology class $\mu_M\in H_n(M;\mathbb{Z}/2)$. Hence for any cohomology class $v\in H^n(M,\mathbb{Z}/2)$, the Kronecker index $v(\mu_M)\in\mathbb{Z}_2$.
My question is, what is the meaning of $v(\mu_M)$? In Hatcher, $v$ needs to act on $n$-dimensional chains with $\mathbb{Z}$ coefficients, it seems that Milnor defines cohomology with coefficients differently. My question is: is there anyway to do this using Hatcher's definition? Or did I miss anything? Thanks in advance!
This is the cap product $v \frown \mu_M \in H_0(M;\Bbb Z/2)$, composed with the point-counting map $c: H_0(M;\Bbb Z/2) \to \Bbb Z/2$. (Rephrased: take $v \frown \mu_M$ in each connected component and add up the results.)
But note that it is also true that $$\text{Hom}_{\Bbb Z}(C_*(X;\Bbb Z), \Bbb Z/2) \cong \text{Hom}_{\Bbb Z/2}(C_*(X;\Bbb Z/2), \Bbb Z/2),$$ so that your concerns about pairing against integer homology classes are unnecessary: this canonical isomorphism shows that you can think of $\Bbb Z/2$-cochains as functions on $\Bbb Z/2$-chains.