Stably diffeomorphic manifolds and l_{2q+1}(Z[\pi])

TL;DR

This paper investigates the algebraic structure of monoids related to stably diffeomorphic manifolds, providing exact sequences and computations that lead to new cancellation results for manifolds with specific fundamental groups.

Contribution

It introduces exact sequences to fully describe l_{2q+1}(Z[]) and computes its Grothendieck group, advancing the understanding of stably diffeomorphic manifolds.

Findings

Exact sequences describing l_{2q+1}(Z[]) as a set

Computation of the Grothendieck group of l_{2q+1}(Z[])

Cancellation results for manifolds with polycyclic-by-finite fundamental groups

Abstract

The monoids l_{2q+1}(Z[\pi]) detect s-cobordisms amongst certain bordisms between stably diffeomorphic 2q-dimensional manifolds and generalise the Wall simple surgery obstruction groups, L_{2q+1}^s(Z[\pi]) \subset l_{2q+1}(Z[\pi]). In this paper we give exact sequences which completely describe l_{2q+1}(Z[\pi]) as a set and which we use to compute its Grothendieck group. As a consequence we deduce cancellation results for stably diffeomorphic manifolds with polycyclic-by-finite fundamental group.