Recent Issues
 The Journal About the Journal Editorial Board Editorial Interests Subscriptions Submission Guidelines Submission Page Policies for Authors Ethics Statement ISSN (electronic): 1472-2739 ISSN (print): 1472-2747 Author Index To Appear Other MSP Journals
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 $\mathrm{SO}\left(3\right)$. 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