Equivariant Surgery Theory for Homology Equivalences under the Gap Condition

Abstract

In the present paper, we discuss an obstruction theory to modify equivariant framed maps on even-dimensional compact smooth manifolds to homology equivalences by equivariant surgery. In 1974, Cappell–Shaneson already developed such obstruction theory in the nonequivariant setting. Our definition of the surgeryobstruction group presents a new aspect of Cappell–Shaneson’s group in the nonequivariant setting and enables us to define directly the surgery obstructions of certain framed maps that are not necessarily connected up to the middle dimension. Using our framework defining the equivariant surgery obstruction, we prove a basic conjecture related to geometric connected sums and algebraic sums of surgery obstructions.