I’ve been struggling lately with Kervaire’s paper “A manifold which does not admit any differentiable structure.” The paper defines the Kervaire invariant of a 4-connected combinatorial 10-manifold and shows that it is automatically zero on smooth manifolds. He then constructs an example of a 4-connected combinatorial 10-manifold whose Kervaire invariant is , concluding that it can admit no smooth structure.

The definition of the Kervaire invariant given in the paper is a little complicated, and I’d like to work through it carefully in this post.

**1. Generalities on framed manifolds**

Let be a framed -dimensional manifold. One of the consequences of being framed is that admits a fundamental class in stable homotopy. Stated more explicitly, there is a *stable* map

which induces an isomorphism in . One can construct such a map by imbedding inside a large euclidean space . The Thom-Pontryagin collapse map then runs

for the normal bundle of in . A choice of trivialization of this normal bundle (that is, a framing) allows us to identify with , and this gives a map

which is an isomorphism in top homology. This gives the desired stable map.

A consequence of this observation is that, in a cell decomposition of , the top cell stably splits off. That is, we can take the stable map and compose it with the crushing map (for instance, identifying with ) such that the composite

is an equivalence. (more…)