Surgery and involutions on 4–manifolds

Abstract

We prove that the canonical 4–dimensional surgery problems can be solved after passing to a double cover. This contrasts the long-standing conjecture about the validity of the topological surgery theorem for arbitrary fundamental groups (without passing to a cover). As a corollary, the surgery conjecture is reformulated in terms of the existence of free involutions on a certain class of 4–manifolds. We consider this question and analyze its relation to the $A,B$–slice problem.