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Homotopy classification of $4$–manifolds whose fundamental group is dihedral

Daniel Kasprowski, John Nicholson and Benjamin Ruppik

Algebraic & Geometric Topology 22 (2022) 2915–2949
Abstract

We show that the homotopy type of a finite oriented Poincaré 4–complex is determined by its quadratic 2–type provided its fundamental group is finite and has a dihedral Sylow 2–subgroup. By combining with results of Hambleton and Kreck and Bauer, this applies in the case of smooth oriented 4–manifolds whose fundamental group is a finite subgroup of SO (3). An important class of examples are elliptic surfaces with finite fundamental group.

Keywords
Whitehead’s Gamma group, homotopy classification of 4–manifolds, Poincaré complexes
Mathematical Subject Classification
Primary: 57K40
Secondary: 16E05, 57N65, 57P10
References
Publication
Received: 7 January 2021
Revised: 1 July 2021
Accepted: 14 July 2021
Published: 13 December 2022
Authors
Daniel Kasprowski
Mathematisches Institut
Rheinische Friedrich-Wilhelms-Universität
Bonn
Germany
John Nicholson
Department of Mathematics
Imperial College London
London
United Kingdom
Benjamin Ruppik
Max-Planck-Institut für Mathematik
Bonn
Germany