Surgery in codimension 3 and the Browder--Livesay invariants

TL;DR

This paper investigates the structure of the inertia subgroup in surgery theory for codimension 3, identifying all possible invariants that obstruct elements from belonging to this subgroup, extending previous approaches.

Contribution

It characterizes all forbidden invariants arising from Browder-Livesay filtrations that determine the inertia subgroup in surgery groups, especially in codimension 3.

Findings

Identifies forbidden invariants in codimensions 0, 1, 2, and 3 for the inertia subgroup.

Generalizes the approach of Hambleton and Kharshiladze to higher codimensions.

Provides a framework for computing the inertia subgroup via Browder-Livesay invariants.

Abstract

The inertia subgroup $I_n(\pi)$ of a surgery obstruction group $L_n(\pi)$ is generated by elements which act trivially on the set of homotopy triangulations $\Cal S(X)$ for some closed topological manifold $X^{n-1}$ with $\pi_1(X)=\pi$. This group is a subgroup of the group $C_n(\pi)$ which consists of the elements which can be realized by normal maps of closed manifolds. In all known cases these groups coincide and the computation of them is one of the basic problems of surgery theory. The computation of the group $C_n(\pi)$ is equivalent to the computation the image of the assembly map $A:H_{n}(B\pi, \bold L_{\bullet})\to L_{n}(\pi)$. Every Browder-Livesay filtration of the manifold $X$ provides a collection of Browder-Livesay invariants which are the forbidden invariants in the closed manifold surgery problem. In the present paper we describe all possible forbidden invariants which can give a Browder-Livesay filtration for computing the inertia subgroup. Our approach is a natural generalization of the approach of Hambleton and Kharshiladze. More precisely, we prove that a Browder-Livesay filtration of a given manifold can give the following forbidden invariants for an element $x\in L_n(\pi_1(X))$ to belong to the subgroup $I_n(\pi)$: the nontrivial Browder-Livesay invariants in codimensions 0, 1, 2 and a nontrivial class of obstructions of a restriction of a normal map to a submanifold in codimension 3.