Motivic to C2-Equivariant Homotopy and Beyond

The problem of classifying mappings of a high-dimensional sphere onto a sphere of lower dimension is a central problem in algebraic topology and has repercussions in geometry and physics. The PI will focus on the context of mappings that preserve special symmetries of the spheres. The research will use recent theoretical developments to advance computational knowledge in this area. This research will be integrated with mentoring activities in the electronic Computational Homotopy Theory (eCHT) community and will be involved in recruiting for activities run by eCHT such as seminars, courses, and networking events for graduate students and postdocs. As an online community, the eCHT increases access to the research community, for example for people with geographical restrictions as well as for those with physical disabilities. The PI and collaborators will leverage motivic/synthetic homotopy theory to perform computations of equivariant stable homotopy groups for the (cyclic) group of order 2, the Klein four group, and the quaternion group of order 8. The main tools to be used are the Bockstein, Adams, and slice spectral sequences. Various techniques will be employed to run these spectral sequences, including the use of Massey products. The PI and collaborators will also work to establish the height 1 Telescope Conjecture in the R-motivic and cyclic-2-equivariant settings, giving a description of certain periodic elements in the corresponding stable homotopy groups. This project is jointly funded by the Topology & Geometric Analysis Program, and the Established Program to Stimulate Competitive Research (EPSCoR). This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.