Homotopy classification of 4–manifolds whose fundamental group is dihedral

Abstract

We show that the homotopy type of an oriented Poincare 4-complex is determined by its quadratic 2-type provided its fundamental group is finite and has a dihedral Sylow 2-subgroup. This applies in the case of smooth oriented 4-manifolds whose fundamental group is a finite subgroup of SO(3), examples of which are elliptic surfaces with finite fundamental group.