The Eta Invariant of Pin Manifolds with Cyclic Fundamental Groups

Abstract

Let ℓ= 2^ν > 1. Let M be an orientable manifold of odd dimension m with $\pi _1 (M) = \mathbb{Z}_\ell$ whose universal cover $\tilde M$ is spin. We define a fixed point free action of $\mathbb{Z}_{2\ell }$ on the product $\tilde M \times \tilde M{\text{ and let }}N: = \tilde M \times \tilde M/\mathbb{Z}_{2\ell } ;{\text{ }}N\left( M \right)$ is non orientable and admits a natural pin^- structure. We express the eta invariant of N(M) in terms of the eta invariant of M and show the map M → N(M) extends to a map of suitably chosen equivariant connective K-theory groups. Let X be a non orientable manifold with $\pi _1 (X) = \mathbb{Z}_{2\ell }$ of even dimension m ≥ 6 whose universal cover is spin. We show that if X admits a metric of positive scalar curvature, then the moduli space of all metrics of positive scalar curvature on X has an infinite number of arc components. If m \n= 2 mod 4 and if w₂(X) = 0, we show X admits a metric of positive scalar curvature if and only if the $\hat A$ genus of the universal cover $\tilde X$ vanishes; this establishes the Gromov-Lawson conjecture in this special case.