The Eta Invariant of Pin Manifolds with Cyclic Fundamental Groups

Let ℓ= 2ν > 1. Let M be an orientable manifold of odd dimension m with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\pi _1 (M) = \mathbb{Z}_\ell $$ \end{document} whose universal cover \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\tilde M$$ \end{document} is spin. We define a fixed point free action of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathbb{Z}_{2\ell } $$ \end{document} on the product \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\tilde M \times \tilde M{\text{ and let }}N: = \tilde M \times \tilde M/\mathbb{Z}_{2\ell } ;{\text{ }}N\left( M \right)$$ \end{document} 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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\pi _1 (X) = \mathbb{Z}_{2\ell } $$ \end{document} 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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\hat A$$ \end{document} genus of the universal cover \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\tilde X$$ \end{document} vanishes; this establishes the Gromov-Lawson conjecture in this special case.