Positive Scalar Curvature for Manifolds with Elementary Abelian Fundamental Group

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 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) with elementary abelian fundamental groups π, except for “toral” classes, and thus our results are automatically applicable once the dimension of the manifold exceeds the rank of π. The proofs involve the detailed structure of BP∗(Bπ), as computed by Johnson and Wilson.