Algebra/Topology Seminar

Speaker: Johannes Ebert

Title: On the rational homology of diffeomorphism groups for some odd-dimensional manifolds

Abstract: I'll talk about new results on the rational cohomology of the classifying space of the diffeomorphism group of the manifold Uⁿ_g,1, which is the connected sum of g copies of Sⁿ x Sⁿ⁺¹, minus a disc. This might be considered as an odd-dimensional analogue of Wⁿ_g,1, the connected sum of g copies of Sⁿ x Sⁿ. The analogous problem for Wⁿ_g,1 was solved in famous work by Madsen--Weiss (n=1) and Galatius--Randal-Williams (n geq 3), for large values of g.

For the manifolds Uⁿ_g,1, we compute the rational cohomology for large g and in degrees up to n-4. The answer looks superficially similar in the sense that the cohomology is an exterior algebra in some generalized Miller--Morita--Mumford classes, with some notable differences.

The proof relies on the surgery-theoretic approach to automorphism groups of manifolds.

This is joint work with Jens Reinhold.