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.