Year of Publication
Doctor of Philosophy (PhD)
Arts and Sciences
Dr. Bertrand Guillou
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 AR(1) of the real motivic Steenrod algebra. The finite subalgebra AR(1), generated by Sq1 and Sq2, of the real motivic Steenrod algebra AR can be given 128 different AR-module structures. We also show that all of these AR-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 AR(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)
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.
Li, Ang, "The v1-Periodic Region in Complex Motivic Ext And a Real Motivic v1-Selfmap" (2022). Theses and Dissertations--Mathematics. 92.