This is a famous problem in algebraic topology which has only recently been settled (except for one case, which is still open) by Hill, Hopkins and Ravenel. The question is about manifolds. It turns out that most framed manifolds are cobordant to a sphere with the usual framing, and the question is the following. For which n is there a framed n-manifold which is not cobordant to Sn? The answer turns out to be n = 2; 6; 14; 30; 62 and possibly 126 (this is the one open case) only.

To solve this problem we need to delve into the world of equivariant stable homotopy theory. The aim of the project is to learn about equivariant stable homotopy theory, and go through as much of the Hill, Hopkins and Ravenel proof as possible.