Benjamin Moldstad (Dept. of Mathematical Sciences, MSU)

04/15/24  4:10pm

Abstract: The unit circle can be regarded as a group via rotations. The goal is to characterize group actions by the circle group in a linear setting. We begin by looking at actions of the circle group on vector spaces. However, in order to account for the topology of the unit circle, the setting of vector spaces is in a sense too restrictive. Therefore, we enrich vector spaces to chain complexes to supply the compelling answer: a circle action on a chain complex is a degree 1 differential. We then consider generalizing this characterization to other linear contexts beyond chain complexes. As I will explain, the above characterization is obstructed precisely by the Hopf map in the more generalized setting. I’ll state a precise theorem articulating this.