PhD defense: Matthias Grey

Title: Rational Homological Stability for Automorphisms of Manifolds

In this thesis we prove rational homological stability for the classifying spaces of the homotopy automorphisms and block diffeomorphisms of iterated connected sums of products of spheres of a certain connectivity.

The results in particular apply to the manifolds

N_g= #_g (S^p x S^q))-int(D^{p+q}), where 2< p< q < 2p-1.

We show that the homology groups

H_i(Baut_\partial (N_g);Q) and  H_i(BBlockdiff_\partial (N_g);Q)

are independent of g for i<g/2-1.

To prove the homological stability for the homotopy automorphisms we show that the groups \pi_1(Baut_\partial (N_g)) satisfy homological stability with coefficients in the homology of the universal covering, which is studied using rational homology theory. The result for the block diffeomorphisms is deduced from the homological stability for the homotopy automorphisms upon using Surgery theory. 

The defense will be in Auditorium 4.