Motivic to C2-Equivariant Homotopy and Beyond (ALN 47.049)

Project Summary Motivic to C2-equivariant homotopy and beyond Overview The PI proposes several projects in R-motivic stable homotopy theory and equivariant stable homotopy theory, for various p-groups. Several of the R-motivic projects have direct consequences in C2-equivariant homotopy theory. Intellectual Merit The computation of the stable homotopy groups of spheres is a central problem in homotopy theory. This problem is now often viewed through the lens of chromatic homo- topy theory, where the homotopy groups are filtered in terms of levels of periodicity. This chromatic filtration on the stable homotopy groups has become a powerful organizational principle. Meanwhile, in the last two decades, the field of motivic stable homotopy the- ory emerged and saw rapid development, while equivariant stable homotopy theory saw an explosion of interest after the Hill-Hopkins-Ravenel solution of the Kervaire invariant problem. The PI proposes work linking these areas of stable homotopy theory. One of the central themes is that motivic homotopy theory over the real numbers R gives an efficient stepping stone towards C2-equivariant homotopy theory. The PI proposes to use recent advances in R-motivic homotopy theory to compute C2-equivariant stable homotopy groups in a range larger than what had been previously computed. These computations then lead to greater understanding of nonequivariant stable homotopy groups, for example via Mahowald’s root invariant. Extending beyond C2-equivariant calculations, the PI has previous work in equivariant stable homotopy theory for the groups K4 and Q8. The PI proposes to use equivariant mo- tivic homotopy theory as well as the equivariant slice filtration to approach computations of those equivariant stable homotopy groups. In previous work with Bhattacharya and Li, the PI began to develop the theory of v1- self-maps in R-motivic and C2-equivariant homotopy theory. This is the beginning of the chromatic filtration in these homotopy theories. The PI proposes to work out an analogue of the Telescope Conjecture in these areas, extending work of Culver and Quigley in the C-motivic case. Broader Impacts The PI will participate in various mentoring activities in the electronic Computational Homotopy Theory community, including being an official mentor for two eCHT NSF- funded postdocs and serving as an administrator for reading courses, graduate student seminars, online courses, and more. The PI is developing a website with Niles Johnson that will display the actions of the R-motivic and C2-equivariant Steenrod algebras on certain standard finite modules.