Representations of groups in the homology of 3-manifolds, and applications

Alex Bartel (Warwick)

Monday 20th February, 2017 15:00-16:00 Maths 204


 If a group G acts on a d-manifold M, then the homology groups H_i(M,Q) are, in a natural way, modules over the group algebra Q[G]. If d is at least 4, then it is easy to see, using the fact that every finitely generated group is the fundamental group of a compact d-manifold, that for every finite group G and every finitely generated Q[G]-module V there exists a compact d-manifold M wih a free action by G such that H_1(M,Q) is isomorphic to V. On the other hand, a consequence of the Riemann-Hurwitz formula is that this is very far from true if d=2. The case d=3 of this "inverse problem" was open until recently, and in this talk I will explain how we have resolved it. I will also sketch an application to isospectral manifolds: for every prime number p, there exist two strongly isospectral hyperbolic 3-manifolds M and M' such that the p-torsion subgroups of H_1(M,Z) and H_1(M',Z) have distinct orders. This result is the natural Riemannian geometry analogue of a statement in number theory whose validity is still an open problem. The techniques are a blend of Riemannian geometry and representation theory of finite groups. This is joint work with Aurel Page.

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