Surgery obstructions on closed manifolds and the Inertia subgroup

TL;DR

This paper proves that for dimensions five and higher, the subgroup of surgery obstructions between closed manifolds coincides with the inertia subgroup, for any finitely-presented group and orientation character.

Contribution

It establishes the equality of the inertia subgroup and the closed manifold subgroup in high dimensions, clarifying their relationship in surgery theory.

Findings

Inertia group and closed manifold subgroup are equal in dimensions ≥6.

The result holds for any finitely-presented group and orientation character.

Provides a geometric understanding of surgery obstructions in high dimensions.

Abstract

The Wall surgery obstruction groups have two interesting geometrically defined subgroups, consisting of the surgery obstructions between closed manifolds, and the inertial elements. We show that the inertia group $I_{n+1}(\pi,w)$ and the closed manifold subgroup $C_{n+1}(\pi,w)$ are equal in dimensions $n+1\geq 6$, for any finitely-presented group $\pi$ and any orientation character $w\colon \pi \to \cy 2$.