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.
Cite this article
Masaharu Morimoto, Equivariant Surgery Theory for Homology Equivalences under the Gap Condition. Publ. Res. Inst. Math. Sci. 42 (2006), no. 2, pp. 481–506DOI 10.2977/PRIMS/1166642112