Homotopy classification of 4-manifolds whose fundamental group is dihedral

TL;DR

This paper establishes that the homotopy type of certain 4-manifolds with dihedral fundamental groups is determined by their quadratic 2-type, advancing classification in geometric topology.

Contribution

It proves that for finite oriented Poincaré 4-complexes with dihedral fundamental groups, the homotopy type is uniquely determined by the quadratic 2-type, extending classification results.

Findings

Homotopy type determined by quadratic 2-type for these 4-manifolds

Applicable to smooth 4-manifolds with finite subgroups of SO(3)

Includes elliptic surfaces with finite fundamental group

Abstract

We show that the homotopy type of a finite oriented Poincar\'{e} 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-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.