Download this book
 Download this article For screen
For printing
Recent Volumes
5: Gauge Theory and Low-Dimensional Topology
4: ANTS XIV
3: Hillman: Poincaré Duality
2: ANTS XIII
1: ANTS X
The Open Book Series
All Volumes
 
About the Series
Ethics Statement
Purchase Printed Copies
Author Index
 
ISSN (electronic): 2329-907X
ISSN (print): 2329-9061
 
MSP Books and Monographs
Other MSP Publications
Poincaré Duality in Dimension 3

Jonathan A. Hillman

Vol. 3 (second edition), 190 pages
 
 
ISBN (print): 978-1-935107-11-8
(electronic) 978-1-935107-12-5
Abstract

Poincaré duality is central to the understanding of manifold topology. Dimension 3 is critical in various respects, being between the known territory of surfaces and the wilderness manifest in dimensions 4. The main thrust of 3-manifold topology for the past half century has been to show that aspherical closed 3-manifolds are determined by their fundamental groups. Relatively little attention has been given to the question of which groups arise. This book is the first comprehensive account of what is known about PD3-complexes, which model the homotopy types of closed 3-manifolds, and PD3-groups, which correspond to aspherical 3-manifolds. In the first half we show that every P2-irreducible PD3-complex is a connected sum of indecomposables, which are either aspherical or have virtually free fundamental group, and largely determine the latter class. The picture is much less complete in the aspherical case. We sketch several possible aproaches for tackling the central question, whether every PD3-group is a 3-manifold group, and then explore properties of subgroups of PD3-groups, unifying many results of 3-manifold topology. We conclude with an appendix listing over 60 questions. Our general approach is to prove most assertions which are specifically about Poincaré duality in dimension 3, but otherwise to cite standard references for the major supporting results.

Target readership: graduate students and mathematicians with an interest in low-dimensional topology.

Milestones
Received: 12 October 2019
First edition: 10 December 2020 (screen/print PDF)
Second edition: 20 January 2024
Authors
Jonathan A. Hillman
The University of Sydney