eCHT Reading Seminar On Group Actions on Exotic Spheres Winter 2026

Summary

Understanding smooth group actions on exotic spheres is a longstanding problem in geometric topology, with interesting connections to algebraic topology and differential geometry. While standard spheres are known to admit faithful actions by relatively high-dimensional compact Lie groups (i.e., SO(n+1) acting on the n-sphere by rotation), some exotic spheres are known not to admit smooth faithful S^3-actions. Similarly, all odd-dimensional standard spheres admit smooth free circle actions, but many odd-dimensional exotic spheres do not.

In this reading group, we will read some classical and recent papers on this subject. Some classical topics we will cover are the stable homotopy classification of exotic spheres, (equivariant) smoothing theory, Smith theory, the classification of free circle actions, and upper bounds on degrees of symmetry. We also hope to cover some more recent topics, such as the use of Mahowald invariants, topological modular forms, and equivariant and chromatic homotopy theory to detect and obstruct actions.

Participant expectations

Participants will take turns giving presentations. The seminar organizers will make sure that presenters have written source material upon which to base their presentations and notes. Participants are expected to attend regularly.

Participants are encouraged to skim the source material being presented each week. They are also encouraged to participate in the private shared online workspace.

As a general rule, students are expected to obtain 1-2 independent study credits from their home universities for their participation in the seminar. This requirement is waived on an individual basis for students whose home universities do not have formal independent study opportunities.

Organizers

The seminar is organized by J.D. Quigley and Oliver Wang . They are assisted by graduate student assistants to be named.

Application

Please submit the application form to apply. Applications received by January 2 will receive full consideration. Contact Dan Isaksen for inquiries about applications.

Schedule

The seminar meets on Tuesdays at 1:00pm eastern time, starting 13 January and ending 28 April, with no meeting on 3 March. Applicants should tentatively reserve these times in their schedules.