Exotic almost complex circle actions on 6-manifolds

TL;DR

This paper constructs a new exotic almost complex circle action on a product of spheres, demonstrating its uniqueness and expanding the classification of 6-manifolds with such symmetries.

Contribution

It shows that one of the previously unknown cases in Jang's classification can be realized via Kustarev's surgery, producing a non-linear, exotic circle action on S^4 x S^2.

Findings

Constructed an exotic almost complex circle action on S^4 x S^2

Proved the action is not equivariantly diffeomorphic to a linear one

Established a uniqueness result for the almost complex structures from Kustarev's construction

Abstract

Jang has proven a remarkable classification of $6$-dimensional manifolds having an almost complex circle action with $4$ fixed points. Jang classifies the weights and associated multigraph into six cases, leaving the existence of connected manifolds fitting into three of the cases unknown. We show that one of the unknown cases may be constructed by a surgery construction of Kustarev, and the underlying manifold is diffeomorphic to $S^4 \times S^2$. We show that the action is not equivariantly diffeomorphic to a linear one, thus giving a new exotic $S^1$-action of on a product of spheres that preserves an almost complex structure. We also prove a uniqueness statement for the almost complex structures produced by Kustarev's construction and prove some topological applications of Jang's classification.