Current Projects
Current Projects
Equivariant Steenrod Operations
joint with Prasit Bhattacharya, Mingcong Zeng, Foling Zou
We develop a general framework for defining Steenrod operations associated to any equivariant cohomology theory represented by a structured genuine G ring spectrum. As an application, we construct two infinite families of nonzero operations for all finite groups.
Status: Submitted
Immersions of C2 Equivariant Projective Spaces via KR theory
joint with Jackson Morris, Manyi Guo, Albert Yang
We compute the Atiyah Real K-theory of C2-equivariant projective spaces and construct immersions of such spaces into multiples of the regular representation. These computations are made tractable by the recent geometric filtration of equivariant projective spaces due to Bhattacharya-Waugh-Zeng-Zou, together with a variant of the localized slice spectral sequence introduced by Meier-Shi-Zeng. As an immediate corollary of these computations, we obtain an equivariant analogue of James periodicity.
Status: Submitted
Equivariant Dyer-Lashof Operations
joint with Prasit Bhattacharya
In this paper we develop a general framework for defining equivariant power operations for all finite groups. Using this framework, we study how these operations interact with HHR norms, additive transfers, restriction maps, and geometric fixed points. When restricted to operations which act on the mod p Bredon homology of "nice" spaces, we also develop Cartan formulas and Adem relations by studying the homology of certain equivariant classifying spaces. This framework extends previous work with Bhattacharya-W-Zeng-Zou for Steenrod operations.
Homology of Equivariant Iterated Loop Spaces
In this paper, we utilize the framework of equivariant Dyer-Lashof operations of Bhattacharya-W to study the homology of free algebras over the terminal N_infty G-operad. This generalizes the nonequivariant computation of J.P. May completed in the 70s. We then explore equivariant generalizations of the nonequivariant computation.
Equivariant Stiefel--Whitney Classes
In this paper, we use previous work of Bhattacharya--Waugh--Zeng--Zou to on Steenrod operations for Bredon cohomology to produce and study properties of G-equivariant Stiefel--Whitney classes and their properties, where G is any finite group. As an example, we compute the Bredon cohomology of B_C2 O(2) with constant F_2 Mackey functor coefficients.