##### Department of Mathematics,

University of California San Diego

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### Math 292 - Topology Seminar

## David Baraglia

#### University of Adelaide

## Non-trivial smooth families of K3 surfaces

##### Abstract:

Let X be a compact, smooth manifold and Diff(X) the diffeomorphism group. The topology of Diff(X) and of the classifying space BDiff(X) are of great interest. For instance, the k-th homotopy group of BDiff(X) corresponds to smooth families over the k-sphere with fibres diffeomorphic to X. By a recent result of Bustamante, Krannich and Kupers, if X has even dimension not equal to 4 and finite fundamental group, then the homotopy groups of BDiff(X) are all finitely generated. In contrast we will show that when X is a K3 surface, the second homotopy group of BDiff(X) contains a free abelian group of countably infinite rank as a direct summand. Our families are constructed using the moduli space of Einstein metrics on K3. Their non-triviality is detected using families Seiberg--Witten invariants.

Host: Jianfeng Lin

### May 5, 2021

### 4:00 PM

### Zoom Meeting ID: 933 6734 4286 Password: topology

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