Let M be a topological spherical space form, i.e. a smooth manifold whose universal cover is a homotopy sphere. We determine the number of path components of the space and moduli space of Riemannian metrics with positive scalar curvature on M if the dimension of M is at least 5 and M is not simply-connected.

We characterize cohomogeneity one manifolds and homogeneous spaces with a compact Lie group action admitting an invariant metric with positive scalar curvature.M S C ( 2 0 2 0 ) 53C20, 57S15 (primary) MAIN RESULTSWhether a given smooth manifold admits a complete Riemannian metric of positive scalar curvature is a long-standing problem in Riemannian geometry. For closed (that is, compact and without boundary) simply-connected manifolds of dimension at least 5, this question has been answered by and Stolz [31]. For non-simply-connected manifolds, however, the problem is still open in many cases (see, for example, the surveys [33,34] by Walsh). Under symmetry assumptions, Lawson and Yau [22] showed that any closed smooth manifold 𝑀 with a smooth (effective) action of a connected, compact, non-abelian Lie group 𝐺 supports an invariant Riemannian metric of positive scalar curvature. Further existence results for manifolds with circle actions have been obtained by Hanke [19] and Wiemeler [35]. Note that the orbit space of a smooth effective circle action on an 𝑛-manifold, 𝑛 ⩾ 1, has dimension 𝑛 − 1. Thus, one generally thinks of manifolds with circle actions as having high-dimensional orbit spaces. In this note, we consider the opposite situation, namely, manifolds with compact Lie group actions whose orbit space is zero-or one-dimensional, and characterize such manifolds admitting positive scalar curvature.