The v₁-Periodic Region in Complex Motivic Ext And a Real Motivic v₁-Selfmap

Author

Ang Li, University of KentuckyFollow

Date Available

5-23-2022

Year of Publication

2022

Document Type

Doctoral Dissertation

Degree Name

Doctor of Philosophy (PhD)

College

Arts and Sciences

Department/School/Program

Mathematics

Advisor

Dr. Bertrand Guillou

Abstract

My thesis work consists of two main projects with some connections. In the first project we establish a v₁ periodicity theorem in Ext over the complex motivic Steenrod algebra. The element h₁ of Ext, which detects the homotopy class \eta in the motivic Adams spectral sequence, is non-nilpotent and therefore generates h₁-towers. Our result is that, apart from these h₁-towers, v₁ periodicity operators give isomorphisms in a range near the top of the Adams chart. This result generalizes well-known classical behavior.

In the second project we consider a nontrivial action of C₂ on the type 1 spectrum Y, which is well-known for admitting a v₁-selfmap. The resultant finite C₂-equivariant spectrum can also be viewed as the complex points of a finite real motivic spectrum. We show that one of the $v₁-selfmaps of Y can be lifted to a selfmap in the real motivic case. Further, the cofiber of the real motivic selfmap is a realization of the subalgebra A^R(1) of the real motivic Steenrod algebra. The finite subalgebra A^R(1), generated by Sq¹ and Sq², of the real motivic Steenrod algebra A^R can be given 128 different A^R-module structures. We also show that all of these A^R-modules can be realized as the cohomology of a 2-local finite real motivic spectrum. The realization results are obtained using an real motivic analogue of the Toda realization theorem. We notice that each realization of A^R(1) can be expressed as a cofiber of a real motivic v₁-selfmap. The C₂-equivariant analogue of the above results then follows because of the Betti realization functor. We identify a relationship between the RO(C₂)-graded Steenrod operations on a C₂-equivariant space and the classical Steenrod operations on both its underlying space and its fixed-points. The second project is joint work with Prasit Bhattacharya and Bertrand Guillou.

Digital Object Identifier (DOI)

https://doi.org/10.13023/etd.2022.186

Funding Information

Part of this study was supported by National Science Foundation grants Division of Mathematical Science (no.:1710379) in 2017 and Division of Mathematical Science (no.:2003204) in 2020.

Recommended Citation

Li, Ang, "The v₁-Periodic Region in Complex Motivic Ext And a Real Motivic v₁-Selfmap" (2022). Theses and Dissertations--Mathematics. 92.
https://uknowledge.uky.edu/math_etds/92