The eta invariant and the Gromov-Lawson conjecture for elementary Abelian groups of odd order

Let M be a compact connected spin manifold of dimension m ⩾ 5. Assume the fundamental group of M is an elementary Abelian p group of rank k where p is an odd prime. If k = 2 and m is arbitrary or if k = 3 and m is odd, we use the eta invariant to show that M admits a metric of positive scalar curvature if and only if the Â-roof genus of M vanishes. This establishes the Gromov-Lawson conjecture for these cases.