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We show that for an orientable non-spin manifold with fundamental group$\mathbb{Z}_2$ and universal cover $S^2\times S^3,$ the moduli space of metricsof nonnegative sectional curvature has infinitely many path components. Therepresentatives of the components are quotients of the standard metric on$S^3\times S^3$ or metrics on Brieskorn varieties previously constructed usingcohomogeneity one actions. The components are distinguished using the relative$\eta$ invariant of the spin$^c$ Dirac operator computed by means of aLefschetz fixed point theorem.

The moduli Space of nonnegatively curved metrics on quotients of $S^2\times S^3$ by involutions