October 4, 2012

The Kervaire invariant II

Posted by Akhil Mathew under topology | Tags: , , |
Leave a Comment 

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…)

 

March 9, 2012

On manifolds homeomorphic to the 7-sphere

Posted by Akhil Mathew under topology | Tags: , , , |
Leave a Comment 

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…)