🔢Algebraic K-Theory Unit 7 – The Bott Periodicity Theorem

Key Concepts and Definitions

• Algebraic K-theory studies algebraic objects (rings, modules, etc.) using techniques from topology and homotopy theory
• Bott periodicity establishes a periodic relationship between the homotopy groups of the stable unitary and orthogonal groups
• The stable homotopy groups of the unitary group $U$ are given by $\pi_{i}(U) \cong \pi_{i+2}(U)$ for all $i \geq 0$
• The stable homotopy groups of the orthogonal group $O$ are given by $\pi_{i}(O) \cong \pi_{i+8}(O)$ for all $i \geq 0$
• The classifying space $BU$ of the stable unitary group is an infinite loop space, with deloopings given by the spaces $\mathbb{Z} \times BU$
• The classifying space $BO$ of the stable orthogonal group is an infinite loop space, with deloopings given by the spaces $\mathbb{Z} \times BO$
• The periodicity isomorphisms are induced by maps $\Omega^{2}BU \to BU$ and $\Omega^{8}BO \to BO$, where $\Omega$ denotes the loop space functor

Historical Context and Development

• Bott periodicity was first discovered by Raoul Bott in the late 1950s while studying the homotopy groups of Lie groups
• Bott's original proof used Morse theory and the study of geodesics on symmetric spaces
• Subsequent proofs were given by Atiyah and Bott using K-theory and the index theorem for elliptic operators
• The theorem was later generalized to other contexts, such as complex and real K-theory, and played a crucial role in the development of algebraic K-theory
• Quillen's work on higher algebraic K-theory in the 1970s relied heavily on the ideas and techniques of Bott periodicity
  • Quillen defined higher K-groups using the plus construction and established their connection to the homotopy theory of the general linear group
• The Bott periodicity theorem has since found applications in various areas of mathematics, including topology, geometry, and mathematical physics

Topological Foundations

• The theorem is formulated in the context of stable homotopy theory, which studies the behavior of topological spaces and maps up to homotopy equivalence
• The unitary group $U(n)$ consists of $n \times n$ complex matrices $A$ satisfying $AA^{*} = I$, where $A^{*}$ is the conjugate transpose of $A$
• The orthogonal group $O(n)$ consists of $n \times n$ real matrices $A$ satisfying $AA^{T} = I$, where $A^{T}$ is the transpose of $A$
• The stable unitary group $U$ is defined as the direct limit of the sequence $U(1) \to U(2) \to U(3) \to \cdots$, where the inclusions are given by $A \mapsto \begin{pmatrix} A & 0 \\ 0 & 1 \end{pmatrix}$
• The stable orthogonal group $O$ is defined similarly as the direct limit of the sequence $O(1) \to O(2) \to O(3) \to \cdots$
• The classifying spaces $BU(n)$ and $BO(n)$ are obtained by taking the quotient of a contractible space by a free action of the respective group
  • These spaces classify principal $U(n)$-bundles and $O(n)$-bundles, respectively
• The stable classifying spaces $BU$ and $BO$ are obtained as the direct limits of the sequences $BU(1) \to BU(2) \to BU(3) \to \cdots$ and $BO(1) \to BO(2) \to BO(3) \to \cdots$

Statement of the Bott Periodicity Theorem

• The Bott periodicity theorem states that there are homotopy equivalences $\Omega^{2}BU \simeq \mathbb{Z} \times BU$ and $\Omega^{8}BO \simeq \mathbb{Z} \times BO$
• These homotopy equivalences induce isomorphisms of homotopy groups $\pi_{i}(BU) \cong \pi_{i+2}(BU)$ and $\pi_{i}(BO) \cong \pi_{i+8}(BO)$ for all $i \geq 0$
• The periodicity can also be stated in terms of the homotopy groups of the stable unitary and orthogonal groups: $\pi_{i}(U) \cong \pi_{i+2}(U)$ and $\pi_{i}(O) \cong \pi_{i+8}(O)$ for all $i \geq 0$
• The theorem implies that the homotopy groups of $BU$ and $BO$ are periodic with periods 2 and 8, respectively
  • Explicitly, $\pi_{i}(BU) \cong \begin{cases} \mathbb{Z} & \text{if } i \text{ is even} \\ 0 & \text{if } i \text{ is odd} \end{cases}$ and $\pi_{i}(BO) \cong \begin{cases} \mathbb{Z} & \text{if } i \equiv 0 \pmod{4} \\ \mathbb{Z}/2\mathbb{Z} & \text{if } i \equiv 1, 2 \pmod{8} \\ 0 & \text{otherwise} \end{cases}$
• The periodicity isomorphisms are compatible with the H-space structures on $BU$ and $BO$, making them infinite loop spaces

Proof Techniques and Strategies

• Bott's original proof used Morse theory to study the topology of the loop spaces of symmetric spaces associated with the unitary and orthogonal groups
• Atiyah and Bott's proof relied on the computation of the K-theory of the classifying spaces $BU$ and $BO$ using the index theorem for elliptic operators
  • They showed that the K-theory of these spaces exhibits the same periodic behavior as the homotopy groups
• Quillen's proof used the plus construction to define higher algebraic K-theory and established a connection between the K-theory of a ring $R$ and the homotopy theory of the general linear group $GL(R)$
• Adams' proof used the methods of stable homotopy theory and the Adams spectral sequence to compute the homotopy groups of $BU$ and $BO$
• Other proofs have been given using techniques from algebraic topology, such as the Atiyah-Hirzebruch spectral sequence and the Adams-Novikov spectral sequence
• The proofs often involve the construction of explicit maps realizing the periodicity isomorphisms and the study of their properties
• The use of cohomology theories, such as K-theory and complex cobordism, plays a crucial role in many of the proofs

Applications in Algebraic K-Theory

• Bott periodicity has been a key tool in the development of algebraic K-theory, which studies the K-theory of rings and other algebraic objects
• Quillen's definition of higher algebraic K-theory uses the plus construction, which is closely related to the Bott periodicity theorem
  • The plus construction can be seen as a generalization of the Bott periodicity maps to the context of the general linear group $GL(R)$ of a ring $R$
• The periodicity of the K-groups of a ring $R$ can be deduced from the Bott periodicity theorem and the connection between algebraic K-theory and the homotopy theory of $BGL(R)^{+}$
• The Bott periodicity theorem has been used to compute the algebraic K-theory of various classes of rings, such as fields, local rings, and rings of integers in number fields
• The theorem has also been applied to the study of the relationship between algebraic K-theory and other invariants, such as cyclic homology and étale cohomology
• Bott periodicity has played a role in the proof of important results in algebraic K-theory, such as the Quillen-Lichtenbaum conjecture and the Bloch-Kato conjecture

Related Theorems and Generalizations

• The Bott periodicity theorem has been generalized to other contexts, such as complex and real K-theory, where similar periodic behavior is observed
• In topological K-theory, the Atiyah-Bott-Shapiro construction provides a geometric realization of the Bott periodicity isomorphisms using Clifford modules and Dirac operators
• The periodicity theorem has an analog in the context of stable homotopy theory, known as the periodicity theorem for the stable homotopy groups of spheres
  • This theorem states that there are certain periodic patterns in the stable homotopy groups of spheres, with periodicities related to the Hopf invariant one problem and the Kervaire invariant one problem
• Bott periodicity has been generalized to the setting of infinite-dimensional Lie groups and their classifying spaces, such as the loop groups and the Kac-Moody groups
• The theorem has also been studied in the context of motivic homotopy theory, where analogues of the Bott periodicity isomorphisms have been constructed for the motivic stable homotopy category
• Generalizations of Bott periodicity have been investigated in the framework of structured ring spectra, such as $E_{\infty}$ ring spectra and $S$-algebras

Exercises and Problem-Solving Approaches

• To develop a deep understanding of the Bott periodicity theorem, it is essential to work through various exercises and problems related to the concepts and techniques involved
• Practice computing the homotopy groups of the unitary and orthogonal groups, as well as their classifying spaces, using the periodicity isomorphisms
  • For example, show that $\pi_{3}(U) \cong \pi_{5}(U) \cong \mathbb{Z}$ and $\pi_{2}(O) \cong \pi_{10}(O) \cong \mathbb{Z}/2\mathbb{Z}$
• Work through the details of the different proof strategies, such as Bott's original proof using Morse theory or Atiyah and Bott's proof using K-theory and the index theorem
  • Try to understand the key ideas and techniques used in each proof and how they relate to the statement of the theorem
• Explore the connections between Bott periodicity and other areas of mathematics, such as algebraic K-theory, stable homotopy theory, and motivic homotopy theory
  • Solve problems that illustrate how the periodicity theorem can be applied to compute invariants or deduce properties in these contexts
• Investigate the generalizations of Bott periodicity to other settings, such as complex and real K-theory or infinite-dimensional Lie groups
  • Work through the construction of the periodicity isomorphisms in these settings and understand how they relate to the classical Bott periodicity theorem
• Develop a robust problem-solving toolkit by mastering the techniques and strategies used in the proofs of the Bott periodicity theorem, such as spectral sequences, characteristic classes, and index theory
  • Apply these tools to solve problems in related areas of algebraic topology and homotopy theory