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 -connected -dimensional manifold, for . 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 is a quadratic refinement**

Let’s next check that the form 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 , we want

In particular, this implies that descends to a function on , as it shows that of an even class is zero in . The associated quotient map is, strictly speaking, the quadratic refinement.

In order to do this, let’s fix . As we saw last time, these can be obtained from maps

by pulling back the generator in degree . Let be a map associated to , and let be a map associated to . We then have that

for the generator of . As we saw, this was equivalent to the definition given last time. (more…)