About this Event
Mike Miller Eismeier, from Columbia University, will present a colloquium titled “Instantons and Dehn Surgery” as part of the Mathematics Department Colloquium Lecture series.
The abstract for the lecture is: There exist many integer homology spheres which are not Dehn surgery on any knot, the first examples being due to Gordon--Luecke. A longstanding question asks whether there exist integer homology spheres Y_n which are not Dehn surgery on any link of fewer than n components; no examples for n > 2 have appeared in the literature.
I will discuss forthcoming work with Ali Daemi which establishes many examples of such Y_n, including hyperbolic examples. The argument relies on formal properties of Froyshov's homology cobordism invariant q_3, in particular that q_3(Y) gives information about the intersection form of any 4-manifold W (whose homology has no 2-torsion) with boundary Y, even indefinite W; previously known invariants only give information when the intersection form of W is definite. The key technical point is the definition of an algebraic object associated to Y which is designed to support mod-2 relative invariants for manifolds W with boundary Y and with b^+(W) > 0, and an analysis of how this object changes as we increase b^+(W).
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