Geometry and Topology Seminar
Monday, February 24, 2020 at 3:00pm to 4:00pm
Building 2, 2-449
182 MEMORIAL DR, Cambridge, MA 02139
Featured Speaker: Mariano Echeverria (Rutgers)
Title: A Generalization of the Tristram-Levine Knot Signatures as a Singular Furuta-Ohta Invariant for Tori.
Abstract: Given a knot K inside an integer homology sphere Y , the Casson-Lin-Herald invariant can be interpreted as a signed count of conjugacy classes of irreducible representations of the knot complement into SU(2) which map the meridian of the knot to a fixed conjugacy class. It has the interesting feature that it determines the Tristram-Levine signature of the knot associated to the conjugacy class chosen. Turning things around, given a 4-manifold X with the integral homology of S1 × S3, and an embedded torus which is homologically non trivial, we define a signed count of conjugacy classes of irreducible representations of the torus complement into SU(2) which satisfy an analogous fixed conjugacy class condition to the one mentioned above for the knot case. Our count recovers the Casson-Lin-Herald invariant of the knot in the product case, thus it can be regarded as implicitly defining a Tristram-Levine signature for tori. This count can also be considered as a singular Furuta-Ohta invariant, and it is a special case of a larger family of Donaldson invariants which we also define.