- The Arf invariant is a nonsingular quadratic form over a field of characteristic 2.
The form that I looked at was: $$ S(q)=|H^1(M^2,\mathbb{Z}_2)|^{-1/2} \sum_{x\in H^1(M^2,\mathbb{Z}_2)} \exp[\pi \;i\; q(x)] $$ see Kirby-Taylor, Pin structures on low-dimensional manifolds.
The $M^2$ is an oriented 2 dimensional manifold with spin structures that has $\mathbb{Z}_2$ valued quadratic forms on $H_1(M^2,\mathbb{Z}_2)$, which obeys $$q(x + y) = q(x) + x ∩ y + q(y) \mod 2,$$ here $x ∩ y$ denotes the $\mathbb{Z}_2$ intersection pairing. The bordism invariant is the Arf invariant.
- Stiefel-Whitney class is $\mathbb{Z}_2$-characteristic class associated to real vector bundles. For example, we can take the $w_i(TM)$ for the tangent bundle of the base manifold $M$.
Question:
Is there some precise way to pair the Arf invariant $S(q)$ with Stiefel-Whitney class $w_i(TM)$? How would one write and define them formally as a topological invariant? Like $$ S(q) \cup w_{d-2}(TM)? $$ where the $M$ is a $d$-dimensional manifold? How do one cup product this two objects formally?
Must this $S(q) \cup w_{d-2}(TM)$ be a $\mathbb{Z}_2$-characteristic class?
What will be a nontrivial manifold generator $M$ for such an invariant $S(q) \cup w_{d-2}(TM)$?