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Johannes Ebert: On the homology of the diffeomorphism group of some odd-dimensional manifolds

Time: Tue 2022-09-27 10.00

Location: Zoom

Video link: Meeting ID: 671 9782 8265

Lecturer: Johannes Ebert (Universität Münster)

Abstract: 

I’ll talk about new results on the rational cohomology of the classifying space of \(\text{Diff}(U^n_{g,1})\), where \(U^n_{g,1}\) denotes the connected sum of g copies of \(S^n\times S^{n+1}\), minus a disc. We consider these manifolds as an odd-dimensional analogue of \(W^n_{g,1}\), the connected sum of g copies of \(S^n\times S^n\), which features prominently in well-known work by Madsen–Weiss and Galatius–Randal-Williams.

We compute the rational cohomology of \(\text{BDiff}_{\partial}(U^n_{g,1} )\) for large g and in degrees up to n − 4. The answer looks superficially similar to the even-dimensional case in the sense that the cohomology is an exterior algebra in some generalized Miller–Morita–Mumford classes, with some notable differences.
The computation is in two steps. The first is to compute the cohomology of the block diffeomorphism space \(B\widetilde{\text{Diff}} _{\partial}(U^n_{g,1} )\), and the second one to compare this withthe actual diffeomorphism groups.

I will focus on the first step in the talk.

This is joint work with Jens Reinhold.