The Gromov-Lawson-Rosenberg conjecture for some finite groups

TL;DR

This paper proves the Gromov-Lawson-Rosenberg conjecture for certain finite groups by calculating topological obstructions and constructing manifolds with positive scalar curvature.

Contribution

It extends the conjecture's validity to specific finite groups, including the Klein 4-group, dihedral groups, semi-dihedral group, and (Z/2)^3.

Findings

Proved the conjecture for Klein 4-group, dihedral groups, semi-dihedral group, and (Z/2)^3.

Calculated the connective real homology ko_*(BG) for these groups.

Constructed manifolds with positive scalar curvature explicitly.

Abstract

The Gromov-Lawson-Rosenberg conjecture for a group G states that a compact spin manifold with fundamental group G admits a metric of positive scalar curvature if and only if a certain topological obstruction vanishes. It is known to be true for G=1, if G has periodic cohomology, and if G is a free group, free abelian group, or the fundamental group of an orientable surface. It is also known to be false for a large class of infinite groups. However, there are no known counterexamples for finite groups. In this dissertation we will give a general outline of the positive scalar curvature problem, and sketch proofs of some of the known positive and negative results. We will then focus on finite groups, and proceed to prove the conjecture for the Klein 4-group, all dihedral groups (joint with Michael Joachim), the semi-dihedral group of order 16 (joint with Kijti Rodtes), and the rank three group (Z/2)^3. The topological obstruction in question lies in the connective real homology ko_*(BG) of the classifying space of BG. Our method of proof is to first sketch calculations of ko_*(BG), using the techniques and calculations of Bruner and Greenlees. We then give explicit geometric constructions to produce sufficiently many manifolds of positive scalar curvature.