Corks, involutions, and Heegaard Floer homology

Abstract

Building on the algebraic framework developed by Hendricks, Manolescu, and Zemke, we introduce and study a set of Floer-theoretic invariants aimed at detecting corks. Our invariants obstruct the extension of a given involution over any homology ball, rather than a particular contractible manifold. Unlike previous approaches, we do not utilize any closed 4-manifold topology or contact topology. Instead, we adapt the formalism of local equivalence coming from involutive Heegaard Floer homology. As an application, we define a modification \Theta_\mathbb{Z}^{\tau} of the homology cobordism group which takes into account an involution on each homology sphere, and prove that this admits a \mathbb{Z}^\infty -subgroup of strongly nonextendable corks. The group \Theta_\mathbb{Z}^{\tau} can also be viewed as a refinement of the bordism group of diffeomorphisms. Using our invariants, we furthermore establish several new families of corks and prove that various known examples are strongly nonextendable. Our main computational tool is a monotonicity theorem which constrains the behavior of our invariants under equivariant negative-definite cobordisms, and an explicit method of constructing such cobordisms via equivariant surgery.

The $(2,1)$-cable of the figure-eight knot is not smoothly slice

“…The homotopy type of the pair (CF − (Y, s), ι) is a diffeomorphism invariant of (Y, s); ι should be thought of as an additional internal symmetry of CF − (Y, s). The importance of considering ι in addition to τ was observed in [DHM22]; in the current article, the usage of ι will likewise be essential.…”

“…If K has slice disk D, then such an extension is obtained by taking τ W to be the branching action over D. Thus, asking for the existence of an equivariant homology ball also provides an obstruction to the sliceness of K, which is (in principle) a strictly stronger refinement of the previous. 3 This strategy was employed successfully in several examples by the second author (in joint work with Alfieri and Stipsicz [AKS20]) and the first and third authors (in joint work with Hedden [DHM22]); see also [BH21,KMT21].…”

“…In both [AKS20] and [DHM22], the obstruction to the existence of such a homology ball was established through the use of Heegaard Floer homology. We recall the essential features of this formalism here.…”

“…There is a slight subtlety in that generally speaking, only the based mapping class group acts naturally on CF − (Y, s). However, if Y is a rational homology sphere (as will be the case in our setting), it follows from work of Zemke [Zem15] that this subtlety can be disregarded; see [DHM22,Lemma 4.1]. Finally, we also make essential use of the involutive Heegaard Floer package of Hendricks and Manolescu [HM17].…”

“…The first two parts of Theorem 2.1 are standard results in the literature, while the third is work of Hendricks and Manolescu [HM17]. The only subtlety in establishing the remaining claim is to analyze the choice of path needed to define F W,s (see [Zem15]), since this may not be invariant under τ ; see [DHM22,Section 6]. This is done in the proof of [DHM22, Theorem 1.2] (in fact, for a much wider class of cobordisms).…”