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…)

It has been known since Milnor’s famous paper that two smooth manifolds can be homeomorphic without being diffeomorphic. Milnor showed that certain sphere bundles over ${S^4}$ were homeomorphic but not diffeomorphic to the 7-sphere ${S^7}$. In later papers, Milnor constructed a number of additional examples of exotic spheres.

In this post, I’d like to give a detailed presentation of the argument in Milnor’s first paper.

1. Distinguishing homeomorphic manifolds

Suppose you have a ${4k-1}$-dimensional manifold ${M}$ which is known to be homeomorphic to the sphere ${S^{4k-1}}$. There are a number of criteria for this: for instance, ${M}$ could admit a cover by two charts, or ${M}$ could admit a function with only two critical points. The goal is to prove then that ${M}$ is not diffeomorphic to ${S^{4k-1}}$. Obviously the standard invariants in topology see only homotopy type and are useless at telling apart ${M}$ and ${S^{4k-1}}$. One needs to define an invariant that relies on the smooth structure of ${M}$ in some way.

It can be shown that any such ${M}$ is an oriented boundary, ${M = \partial B}$, for a ${4k}$-manifold ${B}$. This is a deep fact, but in practice, the manifolds ${M}$ come with explicit ${B}$‘s already, and one might as well define the invariant below on boundaries. Milnor’s strategy is to define an invariant in terms of ${B}$ (which will depend very much on the smooth structure on ${M}$), in such a way that it will turn out to not depend on the choice of ${B}$. (more…)