The diffeomorphism group of a 4-manifold

Geometry Topology Seminar
Monday, January 24, 2022 - 2:00pm for
Online (Zoom)
Danny Ruberman – Brandeis University –
Anubhav Mukherjee

Associated to a smooth n-dimensional manifold are two infinite-dimensional groups: the group of homeomorphisms Homeo(M), and the group of diffeomorphisms, Diff(M). For manifolds of dimension greater than 4, the topology of these groups has been intensively studied since the 1950s. For instance, Milnor’s discovery of exotic 7-spheres immediately shows that there are distinct path components of the diffeomorphism group of the 6-sphere that are connected in its homeomorphism group.  The lowest dimension for such classical phenomena is 5. 

I will discuss recent joint work with Dave Auckly about these groups in dimension 4. For each n, we construct a simply connected 4-manifold Z and an infinite subgroup of the nth homotopy group of Diff(Z) that lies in the kernel of the natural map to the corresponding homotopy group of Homeo(Z). These elements are detected by (n+1)—parameter gauge theory. The construction uses a topological technique.  I’ll mention some other applications to embeddings of surfaces and 3-manifolds in 4-manifolds.

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