Corks, involutions, and Heegaard Floer homology

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