The eta invariant of twisted products of even dimensional manifolds whose fundamental group is a cyclic 2 group (Q1566811)

Let \(M\) be an even dimensional manifold with the cyclic fundamental group \(Z_l\) \((l= 2^\nu)\), and assume that the universal cover \(\widetilde{M}\) is spin. The author relates the eta invariant of \(N(M)= \widetilde{M}\times \widetilde{M}/ Z_{2l}\) to the eta invariant of \(M\) (Theorem 1.6). The main theorem of the paper is the following Theorem 5.1: Let \(M\) be a connected closed non-orientable manifold of dimension \(m\) with \(\pi_1(M)= Z_4\). Assume that \(M\) admits a flat \(\text{pin}^c\) structure. (1) If \(m= 4k\geq 8\) and if \(\omega_2 (M)\neq 0\), then \(M\) admits a metric of positive scalar curvature. (2) If \(m= 4k+ 2\geq 6\) and if \(\omega_2(M)= 0\), then \(M\) admits a metric of positive scalar curvature. This result can be viewed as a special case of Gromov-Lawson conjecture in the nonorientable setting.