The Gromov-Lawson-Rosenberg conjecture for some finite groups

• Main Author: Malhotra, Arjun
• Published: University of Sheffield 2011
• Subjects: 512.23
• Online Access: http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.556648

The Gromov-Lmvson-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 obstruct ion vanishes. It is known to be true for G = 1 . if 0 has periodic cohomology, and if 0 is a free group, free abelian group, or the fundamental group of an orient able 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, the semi-dihedral group of order 16, and the rank three group (Z2)3. Throughout the thesis. ko will represent the connective real K-theory spectrum, and KO the periodic real K-theory, and Z2 the field with two elements. It turns out that that the topological obstruction in question lies in the connective real homology ko*(BG) of the classifying space of G. Indeed, 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.