MT 855: Applications of homology-type invariants in low-dimensional topology
Spring 2014
Time/Location: MW 12:00-1:20, McGuinn 526
Instructor: Eli Grigsby
Office: Carney 357
Course description: We will survey the many exciting applications of homology-type invariants-in particular, Heegaard-Floer homology and Khovanov homology-to questions in knot theory and low-dimensional topology. These will probably include (and may not be limited to) bounds on the 4-ball genus of knots, detection of the Seifert genus and fiberedness of knots (along with unknot detection), trivial mapping class detection for surfaces with boundary, and obstructions to the existence of exceptional Dehn surgeries. We will review the constructions of Khovanov and Heegaard-Floer homology, as well as historical background on the problems discussed, as needed.
Useful (for me) Background references:
Glen Bredon, Topology and Geometry
Robert Gompf and András Stipsicz, 4-manifolds and Kirby calculus
Victor Guillemin and Alan Pollack, Differential topology
Allen Hatcher, Algebraic topology
John Milnor and James Stasheff, Characteristic classes
Khovanov homology references:
Mikhail Khovanov, A categorification of the Jones polynomial
Dror Bar-Natan, On Khovanov's categorification of the Jones polynomial
Heegaard Floer homology references:
Peter Ozsváth and Zoltan Szabó, Holomorphic disks and topological invariants for closed three-manifolds
Peter Ozsváth and Zoltan Szabó, Holomorphic disks and knot invariants
Jacob Rasmussen, Floer homology and knot complements (PhD thesis)
Peter Ozsváth and Zoltan Szabó, On Heegaard diagrams and holomorphic disks
Manolescu, Ciprian, An introduction to knot Floer homology
Topic 1: 4-ball genus of knots and the topological Milnor conjecture
Historical background:
Peter Kronheimer and Tomasz Mrowka, Gauge theory for embedded surfaces, I (Introduction)
Lee Rudolph, Some knot theory of complex plane curves
John Milnor, Singular points of complex hypersurfaces
Charles Livingston, A survey of classical knot concordance (Sec. 1-3)
Homology-type invariants and the four-ball genus:
Eun Soo Lee, An endomorphism of the Khovanov invariant
Jacob Rasmussen, Khovanov homology and the slice genus
Ozsváth and Szabó, Knot Floer homology and the four-ball genus
Stanislav Jabuka, Heegaard Floer homology and knot concordance: a survey of recent advances
Ozsváth and Szabó, On knot Floer homology and lens space surgeries
Louise Moser, Elementary surgery along a torus knot