Let M be a 2q-manifold M. The stable class of M is the set of diffeomorphism classes of manifolds which are stably diffeomorphic to M and have the same Euler characteristic as M. (Recall that manifolds M_0 and M_1 are called stably diffeomorphic if they become diffeomorphic after taking connected sums with copies of S^q x S^q.)

While classifying 2q-manifolds up to stable diffeomorphism is a bordism-theoretic problem, exploring inside the stable class involves delicate quadratic algebra. This algebra starts with Wall’s L-groups, continues to Kreck’s l-monoids and is expected to be complicated in general.

This talk presents two new tools for describing the stable class. The first is a realisation result for elements of Kreck’s l-monoids and leads to examples of manifolds with arbitrarily many homotopy types within their stable class. The second is the “Q-form hypothesis”, which proposes an invariant to classify the stable class up to the action of Wall’s L-groups.

The realisation result for l-monoids is part of joint work with Anthony Conway, Mark Powell and Joerg Sixt. The Q-form hypothesis features in joint projects with Csaba Nagy and Daniel Kasprowski.