A question on Kronecker Index

Question

I am reading A book by Milnor (Lectures on characteristic classes) and I can across this section on Stiefel-Whitney numbers (page 16) and he uses the Kronecker index but never defines it (He says to look on page 118 in Lefschetz Algebraic Topology but then he uses some $\beta$ function and says this is from earlier but I can't find it, so that didn't help). Could someone give me the definition he is using? He defines the fundamental class with coefficients in $\mathbb{Z_2}$ and calls this $\mu$. I understand that part I think. But he doesn't define Kronecker index = $<?, \mu>$. Thanks.

Answer 1

I guess you mean the subsection about Stiefel-Whitney numbers. The Kronecker index is nothing more than evaluation! I.e. the evaluation map $H^n(M;\mathbb Z/2) \otimes H_n(M; \mathbb Z/2) \to \mathbb Z/2$ (or rather: induced by the evaluation map on chain level).

You can look at elements $x_i$ in the cohomology ring of the compact manifold, which live in the top cohomology after multiplying(cupping) them all together and consider their kronecker index $\langle x_1 \cup \cdots \cup x_k , \mu \rangle$. This leads to the definition of Stiefel Whitney numbers.