# Homotopy groups of the diffeomorphism group of a 4-manifold

# Homotopy groups of the diffeomorphism group of a 4-manifold

Using multi-parameter Yang-Mills gauge theory, we define homomorphisms from the homotopy groups of Diff(M) to Z for a smooth oriented 4-manifold satisfying some simple homological conditions. Some 20 years ago, I found manifolds M supporting a diffeomorphism that is not smoothly isotopic to the identity (although it is topologically.) In a joint project with Dave Auckly, I show the analogous fact for loops of diffeomorphisms and outline a construction for higher homotopy groups.

Theorem: There is a simply-connected 4-manifold such that the homomorphism defined on the fundamental group of Diff(M) is non-trivial. The corresponding loop of diffeomorphisms is trivial as a loop of homeomorphisms.

The construction of loops of diffeomorphisms makes use of a stabilization theorem for surfaces with Auckly, Kim, and Melvin. The calculation of the invariant relies on a combination of splitting and wall-crossing theorems.