On the classification of fake lens spaces

TL;DR

This paper classifies fake lens spaces of dimension five or higher with cyclic fundamental groups, extending previous results, and explores the suspension map to describe torsion invariants geometrically.

Contribution

It provides a comprehensive classification of fake lens spaces for all cyclic groups and analyzes the suspension map's role in understanding their torsion invariants.

Findings

Classification of fake lens spaces for cyclic groups of order N >= 2

Extension of Wall's results to new cases including N a power of 2

Geometric description of torsion invariants via suspension maps

Abstract

In the first part of the paper we present a classification of fake lens spaces of dimension >= 5 whose fundamental group is the cyclic group of order N >= 2. The classification uses and extends the results of Wall and others in the case N = 2 and N odd and the results of the authors of the present paper in the case N a power of 2. In the second part we study the suspension map between the simple structure sets of lens spaces of different dimensions. As an application we obtain an inductive geometric description of the torsion invariants of fake lens spaces.