7.2 The Bott periodicity theorem for algebraic K-theory

Bott periodicity in algebraic K-theory reveals a surprising pattern: K-groups repeat every two dimensions. This theorem connects algebra and topology, showing that K_n(R) is isomorphic to K_{n+2}(R) for any ring R and n ≥ 0.

This result simplifies calculations of higher K-groups and provides insights into the structure of algebraic K-theory. It's a powerful tool that bridges abstract algebra with topology, showcasing the deep connections between seemingly disparate mathematical fields.

Bott Periodicity Theorem

Statement and Key Concepts

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• Natural isomorphism K_n(R) ≅ K_{n+2}(R) for any ring R and integer n ≥ 0 establishes a periodic pattern in higher algebraic K-groups
• Periodicity of 2 in algebraic K-theory contrasts with periodicity of 2 for complex topological K-theory and 8 for real topological K-theory
• Involves stable homotopy groups of the infinite general linear group GL(R)
• Expressed using BGL(R)^+, the plus construction applied to the classifying space of GL(R)
• Requires understanding of higher algebraic K-groups (K_n for n > 1) and the plus construction in algebraic topology
• Connects abstract algebra and topology through K-theory

Mathematical Formulation

• Formal statement: $K_n(R) \cong K_{n+2}(R)$ for all n ≥ 0
• Homotopy-theoretic formulation: $\pi_i(BGL(R)^+) \cong \pi_{i+2}(BGL(R)^+)$ for i ≥ 1
• Spectrum formulation: K(R) is a 2-periodic spectrum
• Relation to topological K-theory: $K_n(\mathbb{C}) \cong K^{-n}(pt)$ (complex case)
• Real K-theory analog: $[KO_n](https://www.fiveableKeyTerm:ko_n)(\mathbb{R}) \cong KO^{-n}(pt)$ with 8-fold periodicity

Importance of Bott Periodicity

Computational Applications

• Simplifies calculation of higher algebraic K-groups by reducing infinite sequence to repeating pattern
• Extends techniques for computing K_0(R) and K_1(R) to all K_n(R) using periodicity isomorphism
• Enables more effective use of long exact sequences in K-theory by reducing unknown terms
• Facilitates deduction of stability results for general linear group GL(R)
• Applies to K-group calculations for specific rings (fields, number rings, group rings)
• Limitations arise for rings with complex structure or in low dimensions where periodicity may not be apparent

Theoretical Significance

• Bridges abstract algebra and topology through deep connection between algebraic and topological K-theory
• Provides insights into global structure of algebraic K-theory spectra
• Plays crucial role in Quillen-Lichtenbaum conjecture, relating algebraic K-theory to étale cohomology
• Applications extend to algebraic geometry, number theory, and operator algebras
• Simplifies structure of algebraic K-theory, reducing infinite sequence to repeating pattern
• Establishes algebraic K-theory as a cohomology theory with periodicity, similar to topological K-theory

Proof of Bott Periodicity

Key Components and Techniques

• Constructs map between K-theory spaces inducing isomorphisms on homotopy groups
• Utilizes Quillen's plus construction as key ingredient, enabling homotopy-theoretic techniques
• Employs concepts of H-spaces and infinite loop spaces from algebraic topology
• Applies delooping techniques to relate K-theory spaces of different degrees
• Demonstrates certain maps between K-theory spaces are homotopy equivalences
• Involves spectral sequences, particularly the Atiyah-Hirzebruch spectral sequence
• Highlights role of Grothendieck group K_0(R) and its relationship to higher K-groups

Proof Outline

• Step 1: Construct a map f: BGL(R)^+ → Ω^2BGL(R)^+
• Step 2: Show f induces isomorphism on homotopy groups π_i for i ≥ 1
• Step 3: Use Whitehead theorem to conclude f is a homotopy equivalence
• Step 4: Apply delooping to obtain periodicity for all higher K-groups
• Step 5: Utilize spectral sequence arguments to handle low-dimensional cases
• Step 6: Extend result to non-connective K-theory spectrum

Calculating K-groups with Bott Periodicity

Practical Applications

• Reduces higher K-group calculations to lower-dimensional cases (K_0, K_1, K_2)
• Applies to stability theorems for general linear groups (stable range calculations)
• Facilitates computations for specific rings (Z, finite fields, local rings)
• Enhances effectiveness of other K-theory tools (localization sequence, Mayer-Vietoris sequence)
• Limitations arise in low dimensions or for rings with complex structure

Computational Techniques

• Use periodicity to reduce K_n(R) to K_0(R), K_1(R), or K_2(R) depending on n mod 2
• Combine with exact sequences (localization, Mayer-Vietoris) to solve for unknown K-groups
• Apply in conjunction with other K-theory results (e.g., Fundamental Theorem of K-theory)
• Utilize known calculations of K_0 and K_1 (e.g., K_0(Z) = Z, K_1(Z) = Z/2Z) to deduce higher K-groups
• Employ stability results for GL(R) to simplify calculations in stable range