Noncommutative Borsuk-Ulam Theorems

Abstract

OF THE DISSERTATION Noncommutative Borsuk-Ulam Theorems by Benjamin Passer Doctor of Philosophy in Mathematics Washington University in St. Louis, 2016 Professor John McCarthy, Chair Professor Xiang Tang, Co-Chair The Borsuk-Ulam theorem in algebraic topology shows that there are significant restrictions on how any topological sphere S interacts with the antipodal Z2 action of reflection through the origin (x 7→ −x). For example, any map f : S → S which is continuous and odd (f(−x) = −f(x)) must be homotopically nontrivial. We consider various equivalent forms of the theorem in terms of the function algebras C(S) and examine which forms generalize to certain noncommutative Banach and C∗-algebras with finite group actions. Chapter 1 contains background material on C∗-algebras, K-theory, and group actions. Next, in Chapter 2, we examine statements related to the Borsuk-Ulam theorem that may be applied on Banach algebras with Z2 actions; this work indicates when roots of elements do not exist and is motivated by the results of Ali Taghavi in [49]. We see that a variant of the Borsuk-Ulam theorem on C(S) written in terms of individual odd elements of C(S) does not extend to the noncommutative setting. In Chapter 3, we show that antipodally equivariant maps between θ-deformed spheres of the same dimension are nontrivial on K-theory. This generalizes the commutative case and parallels the work of Makoto Yamashita in [55] on the q-spheres, although our methods are quite different. Finally, Chapter 4 concerns a conjecture of Ludwik D ‘ abrowski in [14] that seeks to generalize noncommutative Borsuk-Ulam theory to arbitrary C∗-algebras through the use of unreduced suspensions. We prove D ‘ abrowski’s conjecture and propose a new direction for continued study.