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Series: Geometry Topology Seminar

Series: Geometry Topology Seminar

Given an m-dimensional manifold M that is homotopy equivalent to an n-dimensional manifold N (where n<m), a spine of M is a piecewise-linear embedding of N into M (not necessarily locally flat) realizing the homotopy equivalence. When m-n=2 and m>4, Cappell and Shaneson showed that if M is simply-connected or if m is odd, then it contains a spine. In contrast, I will show that there exist smooth, compact, simply-connected 4-manifolds which are homotopy equivalent to the 2-sphere but do not contain a spine (joint work with Tye Lidman). I will also discuss some related results about PL concordance of knots in homology spheres (joint with Lidman and Jen Hom).

Series: Geometry Topology Seminar

Series: Geometry Topology Seminar

Series: Geometry Topology Seminar

Series: Geometry Topology Seminar

Series: Geometry Topology Seminar

Series: Geometry Topology Seminar

Khovanov homology and knot Floer homology are two knot invariants which are defined using very different techniques, with Khovanov homology having its roots in representation theory and knot Floer homology in symplectic geometry. However, they seem to contain a lot of the same topological data about knots. Rasmussen conjectured that this similarity stems from a spectral sequence from Khovanov homology to knot Floer homology. In this talk I will give a construction of this spectral sequence. The construction utilizes a recently defined knot homology theory HFK_2 which provides a framework in which the two theories can be related.

Series: Geometry Topology Seminar

Series: Geometry Topology Seminar

It is well known that two knots in S^3 are ambiently isotopic if and only if there is an orientation preserving automorphism of S^3 carrying one knot to the other. In ;this talk, we will examine a family of smooth 4-manifolds in which the analogue of this fact does not hold, i.e. each manifold contains a pair of smoothly embedded, homotopic 2-spheres that are related by a diffeomorphism, but are not smoothly isotopic. In particular, the presence of 2-torsion in the fundamental groups of these 4-manifolds can be used to obstruct even a topological isotopy between the 2-spheres; this shows that Gabai's recent ``4D Lightbulb Theorem" does not hold without the 2-torsion hypothesis.