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

TL;DR

This paper demonstrates that certain non-spin 4-manifolds with fundamental group Z2 and universal cover S^2×S^3 have infinitely many distinct classes of nonnegative curvature metrics, distinguished by eta invariants.

Contribution

It establishes the existence of infinitely many path components in the moduli space of nonnegative curvature metrics on specific quotients of S^2×S^3, using eta invariants for distinction.

Findings

Infinitely many path components in the moduli space of nonnegative curvature metrics.

Components distinguished by relative eta invariants of the spin^c Dirac operator.

Metrics include quotients of standard metrics and those on Brieskorn varieties.

Abstract

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 metrics of nonnegative sectional curvature has infinitely many path components. The representatives of the components are quotients of the standard metric on $S^3\times S^3$ or metrics on Brieskorn varieties previously constructed using cohomogeneity one actions. The components are distinguished using the relative $\eta$ invariant of the spin$^c$ Dirac operator computed by means of a Lefschetz fixed point theorem.