Applications of Involutive Heegaard Floer Homology

Abstract

We use Heegaard Floer homology to define an invariant of homology cobordism. This invariant is isomorphic to a summand of the reduced Heegaard Floer homology of a rational homology sphere equipped with a spin structure and is analogous to Stoffregen's connected Seiberg-Witten Floer homology. We use this invariant to study the structure of the homology cobordism group and, along the way, compute the involutive correction termsd and d for certain families of three-manifolds.

Connected Floer homology of covering involutions

“…By Zorn's lemma maximal self-local equivalences always exist. The following lemma summarises the results of [9,Section 3].…”

“…Adapting ideas from [9], the chain complex CF − (Σ(K), s 0 ), equipped with τ # , provides concordance invariants of the knot K as follows. Recall [9, Definition 2.5] regarding ι-complexes:…”

“…If C(v 1 ) contains a pair of leaves {v, Jv} with weight larger than the weight of any leaf in S then we choose one such pair with largest possible weight and we add it to S. By keep iterating this procedure until γ merges with the long stem we end up with a set S of distiguished leaves. We denote by M the smallest graded root M ⊂ R containing S. According to [9,Proposition 7.5] we have that HFB − conn (K) = H − (M, w| M ). The resulting M is the monotone subroot of (R, w).…”

“…This group is interesting only for those spin c structures which originate from a spin structure, and provides a new and rather sensitive diffeomorphism invariant of the underlying spin three-manifold. A further application of the above involution appeared in the work of Hendricks, Hom and Lidman [9], where connected Heegaard Floer homology HF − conn (Y ), a submodule of HF − (Y ) was defined. This submodule turned out to be a homology cobordism invariant.…”

“…Adapting the method of [9] for defining new homology cobordism invariants of rational homology spheres, we define the connected branched Floer homology HFB − conn (K) of a knot K ⊂ S 3 as follows. Consider a sel-local equivalence f max : CF − (Σ(K), s 0 ) → CF − (Σ(K), s 0 ) which commutes (up to homotopy) with τ # and has maximal kernel among such endomorphisms.…”