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 longstanding 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.

Surfaces in 4-manifolds and the surgery conjecture

“…) is a long-standing open problem, one can construct [11] a double cover N of this hypothetical manifold M (and of all other canonical manifolds) -in fact N is a smooth manifold. Therefore the surgery conjecture is equivalent to the existence of a free topological involution on a certain class of 4-manifolds.…”