#### Year of Publication

2022

#### Degree Name

Doctor of Philosophy (PhD)

#### Document Type

Doctoral Dissertation

#### College

Arts and Sciences

#### Department/School/Program

Mathematics

#### First Advisor

Dr. Bertrand Guillou

#### Abstract

My thesis work consists of two main projects with some connections. In the first project we establish a v_{1} periodicity theorem in Ext over the complex motivic Steenrod algebra. The element h_{1} of Ext, which detects the homotopy class \eta in the motivic Adams spectral sequence, is non-nilpotent and therefore generates h_{1}-towers. Our result is that, apart from these h_{1}-towers, v_{1} 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 C_{2} on the type 1 spectrum Y, which is well-known for admitting a v_{1}-selfmap. The resultant finite C_{2}-equivariant spectrum can also be viewed as the complex points of a finite real motivic spectrum. We show that one of the $v_{1}-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 Sq^{1} and Sq^{2}, 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 v_{1}-selfmap. The C_{2}-equivariant analogue of the above results then follows because of the Betti realization functor. We identify a relationship between the RO(C_{2})-graded Steenrod operations on a C_{2}-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 v_{1}-Periodic Region in Complex Motivic Ext And a Real Motivic v_{1}-Selfmap" (2022). *Theses and Dissertations--Mathematics*. 92.

https://uknowledge.uky.edu/math_etds/92