  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.
