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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

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.

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