Mathematics Seminar
Date: Monday, 8 January 2023
Time: 2:00 PM
Venue: Seminar Hall
SU(2) Representations of Three-Manifold groups
Deepraj Bhat
MIT, USA.
08-01-24

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Abstract

By the resolution of the Poincare conjecture in 3 dimensions, we know that the only closed three-manifold with the trivial fundamental group is the three-sphere. In light of it, one can ask the following question: Suppose M is a closed three-manifold with the property that the only representation \pi_1(M) --> SU(2) is the trivial one. Does this imply that \pi_1(M) is trivial? The class of manifolds M for which this question is interesting (and open) are integer homology spheres. We introduce a new strategy to answer this question and use it to reprove a result in this direction: the half-Dehn surgery on any knot K of genus > 1 in S^3 admits an irreducible representation. The proof uses instanton floer homology. I will give a brief introduction to instanton floer homology and sketch the strategy. This is based on work in progress, some jointly with Zhenkun Li, Charles Stine and Fan Ye.