This is the second post in a series on Kervaire’s paper “A manifold which does not admit any differentiable structure.” In the previous post, we described a form on the middle cohomology of a $k-1$-connected $2k$-dimensional manifold, for $k \neq 1, 3, 7$. In this post, we can define the Kervaire invariant of such a framed manifold, by showing that this defines a form. I’ll try to sketch the proof that there is no framed manifold of Kervaire invariant one in dimension 10.

1. The form $q$ is a quadratic refinement

Let’s next check that the form ${q: H^k(M; \mathbb{Z}) \rightarrow \mathbb{Z}/2}$ defined in the previous post (we’ll review the definition here) is actually a quadratic refinement of the cup product. Precisely, this means that for ${x, y \in H^k(M; \mathbb{Z})}$, we want

$\displaystyle q(x+y) - q(x) - q(y) = (x \cup y)[M].$

In particular, this implies that ${q}$ descends to a function on ${H^k(M; \mathbb{Z}/2)}$, as it shows that ${q}$ of an even class is zero in ${\mathbb{Z}/2}$. The associated quotient map ${q: H^k(M; \mathbb{Z}/2) \rightarrow \mathbb{Z}/2}$ is, strictly speaking, the quadratic refinement.

In order to do this, let’s fix ${x, y \in H^k(M; \mathbb{Z}/2)}$. As we saw last time, these can be obtained from maps

$\displaystyle M \rightarrow \Omega \Sigma S^k$

by pulling back the generator in degree ${k}$. Let ${f_x}$ be a map associated to ${x}$, and let ${f_y}$ be a map associated to ${y}$. We then have that

$\displaystyle q(x) = f_x^*(u_{2k}) [M], \quad q(y) = f_y^*(u_{2k})[M],$

for ${u_{2k}}$ the generator of ${H^{2k}(\Omega \Sigma S^k; \mathbb{Z}/2)}$. As we saw, this was equivalent to the definition given last time. (more…)