Representations of knot concordances and homology cobordisms

Representations of knot concordances and homology cobordisms

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Aliakbar Daemi, Washington University

Zoom link:https://princeton.zoom.us/j/453512481?pwd=OHZ5TUJvK2trVVlUVmJLZkhIRHFDUT09

An important topological invariant of closed 3-manifolds is provided by their fundamental groups, which form a special family of finitely presented groups. For instance, any 3-manifold group admits a presentation with deficiency zero, and it is conjectured that the fundamental group of any closed 3-manifold, non-homeomorphic to the 3-dimensional sphere, admits a non-trivial representation into the Lie group SU(2). Similar constraints do not hold for the fundamental groups of 4-manifolds. However, the special family of 4-manifolds given by the complements of knot concordances and homology cobordisms demonstrate some behaviors similar to the case of 3-manifolds. By focusing on a few topological applications, I explain how Yang-Mills gauge theory can be used to obtain information about the fundamental groups of such 4-manifolds. Last week’s talk by Chris Scaduto introduced some tools which will play a key role in my talk. But I will make this talk independent of Scaduto’s talk and review the necessary background from his talk.