Positive scalar curvature for manifolds with elementary abelian fundamental group

Botvinnik, Boris; Rosenberg, Jonathan

doi:10.1090/S0002-9939-04-07762-7

Positive scalar curvature for manifolds with elementary abelian fundamental group
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by Boris Botvinnik and Jonathan Rosenberg PDF

Proc. Amer. Math. Soc. 133 (2005), 545-556 Request permission

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 $KO_*$-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 $\ge 5$) with elementary abelian fundamental groups $\pi$, except for “toral” classes, and thus our results are automatically applicable once the dimension of the manifold exceeds the rank of $\pi$. The proofs involve the detailed structure of $BP_*(B\pi )$, as computed by Johnson and Wilson.

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