Positive scalar curvature for manifolds with elementary abelian fundamental group

Abstract

The statement often called the Gromov-Lawson-Rosenberg Conjecture asserts that a manifold with finite fundamental group should admit a metric of positive scalar curvature except when the K O ∗ KO_* -valued index of some Dirac operator with coefficients in a flat bundle is non-zero. We prove spin and oriented non-spin versions of this statement for manifolds (of dimension ≥ 5 \ge 5 ) with elementary abelian fundamental groups π \pi , except for “toral” classes, and thus our results are automatically applicable once the dimension of the manifold exceeds the rank of π \pi . The proofs involve the detailed structure of B P ∗ ( B π ) BP_*(B\pi ) , as computed by Johnson and Wilson.