Semi-free actions with manifold orbit spaces

TL;DR

This paper classifies certain smooth, semi-free group actions on simply connected manifolds, revealing the topological structure of the manifolds and their quotients, and establishing restrictions on possible actions and manifold types.

Contribution

It provides a classification of simply connected 5-manifolds with semi-free circle actions and explores restrictions on semi-free $S^3$ actions on 8-manifolds, linking fixed-point data to manifold topology.

Findings

Only connected sums of $S^3$-bundles over $S^2$ admit such circle actions.

Betti numbers of 5-manifolds and quotients are related by a simple formula.

Strong topological restrictions exist for semi-free $S^3$ actions on 8-manifolds.

Abstract

In this paper, we study smooth, semi-free actions on closed, smooth, simply connected manifolds, such that the orbit space is a smoothable manifold. We show that the only simply connected $5$-manifolds admitting a smooth, semi-free circle action with fixed-point components of codimension $4$ are connected sums of $S^3$-bundles over $S^2$. Furthermore, the Betti numbers of the $5$-manifolds and of the quotient $4$-manifolds are related by a simple formula involving the number of fixed-point components. We also investigate semi-free $S^3$ actions on simply connected $8$-manifolds with quotient a $5$-manifold and show, in particular, that there are strong restrictions on the topology of the $8$-manifold.