On the Naturality of the Spectral Sequence from Khovanov Homology to Heegaard Floer Homology

J E Grigsby,S M Wehrli

doi:10.1093/imrn/rnq039

Abstract

In [18], Ozsvath-Szabo established an algebraic relationship, in the form of a spectral sequence, between the reduced Khovanov homology of (the mirror of) a link and the Heegaard Floer homology of its double-branched cover. This relationship, extended in [19] and [4], was recast, in [5], as a specific instance of a broader connection between Khovanov- and Heegaard Floer-type homology theories, using a version of Heegaard Floer homology for sutured manifolds developed by Juhasz in [7]. In the present work, we prove the naturality of the spectral sequence under certain elementary operations, using a generalization of Juhasz’s surface decomposition theorem valid for decomposing surfaces geometrically disjoint from an imbedded framed link.

Highlights

In [5], we recast [18], [19], and [4] as specific instances of a broader relationship between Khovanov- and Heegaard Floer-type homology theories, using a version of Heegaard Floer homology for sutured manifolds developed by Juhasz in [7]
In [4] we prove the existence of a spectral sequence from the Khovanov homology of any admissible balanced tangle, T ⊂ D × I, to the sutured Floer homology of Σ(D × I, T )
In [5], we prove the existence of a similar spectral sequence from the Khovanov homology of a link, L, in the product sutured manifold A × I to the sutured Floer homology of Σ(A × I, L)

Summary

Introduction

By counting holomorphic polygons in a particular choice of Heegaard multi-diagram compatible with LT (resp., LL), one obtains a filtered complex, X(LT ) (resp., X(LL)), with an associated link surgeries spectral sequence whose E2 term is an appropriate version of Khovanov homology for T (resp., for L) and whose E∞ term is the sutured Floer homology of Σ(D × I, T ) (resp., of Σ(A × I, L)). “F1 = F2” means that F1 is filtered quasi-isomorphic (Definition 2.6) to F2, and “F1 ≤ F2” means that F1 is filtered quasi-isomorphic to a direct summand of F2 These naturality theorems follow from an extension of work of Juhasz, who proves, in [8], that the Floer homology of a sutured manifold behaves nicely in the presence of admissible decomposing surfaces (Definition 4.1), properly-imbedded surfaces with boundary intersecting the sutures in a controlled fashion. We discuss the relationship between the stacking operation and a generalized version of the Murasugi sum

Definitions and Notation Conventions

Splitting the Link Surgeries Spectral Sequence

Surface decompositions

Naturality of the Spectral Sequence