Periodicity Uncovered: A Deep Dive into Bott's Theorems in K-Theory and Fiber Bundles

TL;DR

The paper compares Bott periodicity in two settings: topological K-theory and the stable homotopy groups of classical groups. It builds from vector bundles and the foundations of $K(X)$ and $\widetilde{K}(X)$ to establish a Bott map $\beta$ giving $\widetilde{K}(X) \cong \widetilde{K}(S^{2}X)$ (a period-2 phenomenon) via the external product with the canonical line bundle on $S^2$. It then connects these ideas to fiber bundles and Stiefel manifolds to derive Bott periodicity for the stable homotopy groups of $O(n)$, $U(n)$, and $Sp(n)$ with periods $8$, $2$, and $8$, respectively. The work highlights a deep interplay between algebraic K-theory and geometric topology, revealing recurring twofold and eightfold periodic structures that organize classical topology.

Abstract

This paper presents a comparison between two versions of Bott Periodicity Theorems: one in topological K-theory and the other in stable homotopy groups of classical groups. It begins with an introduction to K-theory, discussing vector bundles and their role in understanding the algebraic and topological aspects of these spaces. Then the two versions of Bott periodicity, as well as the topological notions necessary to understand them, are further explored. The aim is to illustrate the connections and distinctions between these two theorems, deepening our understanding of their underlying mathematical structures such as topological K-theory and fiber bundles.

Figures (1)

• Figure 1: Raoul Bott

Theorems & Definitions (52)

• Definition 1.1: Vector Bundle
• Example 1.2: Line Bundle
• Example 1.3: Canonical Line Bundle
• Definition 1.4: $\operatorname{Vect}^n(B)$ and $\operatorname{Vect}_{\mathbb{C}}^n(B)$
• Definition 1.5: Pullback Bundle
• Definition 1.6: Stable Isomorphism
• Proposition 1.7
• Definition 1.8: Reduced K-theory Group
• Example 1.9
• Definition 1.10: K-theory Group $K(X)$