Abstract

Currently Heegaard Floer homology has two different extensions to 3-manifolds with boundary: sutured Floer homology (SFH) for balanced sutured manifolds due to myself, and the more recent bordered Floer homology for 3-manifolds with parametrized boundary due to Lipshitz, Ozsvath, and Thurston. In this talk, I will introduce the notion of cobordism between sutured manifolds and explain how such cobordisms induce maps on SFH . For this end, I had to establish certain functoriality properties of Heegaard Floer homology, which then enabled me to fit the theory of SFH in a TQFT type framework.

Lipshitz et. al proved that bordered Floer homology of a given 3-manifold with parametrized boundary can be obtained from SFH and our cobordism maps. Another application is that we can define maps induced on knot Floer homology by decorated knot coboridsms and this yields a new invariant of surfaces embedded in 4-manifolds. Similar knot cobordism maps in the instanton setting were used by Kronheimer and Mrowka to show that Khovanov homology detects the unknot.