🪡K-Theory Unit 3 – Topological K-Theory: Bott Periodicity

Key Concepts and Definitions

• Topological K-theory studies vector bundles over topological spaces and their stable equivalence classes
• K-group $K(X)$ consists of stable equivalence classes of vector bundles over a compact Hausdorff space $X$
  • Addition in $K(X)$ corresponds to the direct sum of vector bundles
  • Multiplication in $K(X)$ corresponds to the tensor product of vector bundles
• Reduced K-theory $\tilde{K}(X)$ is defined as the kernel of the map $K(X) \to K(*)$ induced by the map $X \to *$ to a point
• Bott periodicity establishes a periodic relationship between the K-groups of a space and its suspensions
• Clifford algebras play a crucial role in the proof of Bott periodicity
  • Real Clifford algebras $Cl_{p,q}$ are generated by elements $e_1, \dots, e_n$ with $e_i^2 = -1$ for $i \leq p$ and $e_i^2 = 1$ for $i > p$
  • Complex Clifford algebras $Cl_n$ are generated by elements $e_1, \dots, e_n$ with $e_i^2 = -1$ and $e_ie_j = -e_je_i$ for $i \neq j$
• Fredholm operators are bounded linear operators with finite-dimensional kernel and cokernel, used in the study of K-theory

Historical Context and Development

• K-theory originated in the work of Alexander Grothendieck in the 1950s, who introduced it as a tool in algebraic geometry
• Topological K-theory was developed by Michael Atiyah and Friedrich Hirzebruch in the 1960s
  • They used it to study vector bundles and their characteristic classes
• Raoul Bott proved the periodicity theorem in 1959, establishing a fundamental result in topological K-theory
• Atiyah and Hirzebruch applied K-theory to the study of characteristic classes and the index theorem
• K-theory has since found applications in various areas of mathematics, including topology, geometry, and mathematical physics
  • It has been used to study the classification of vector bundles, the Atiyah-Singer index theorem, and the structure of C*-algebras
• The development of K-theory has led to the creation of related theories, such as KK-theory and twisted K-theory
• K-theory has also been generalized to other settings, such as equivariant K-theory and K-theory for operator algebras

Foundations of Topological K-Theory

• Topological K-theory studies vector bundles over topological spaces up to stable equivalence
• The K-group $K(X)$ is defined as the Grothendieck group of the monoid of isomorphism classes of vector bundles over $X$ under the direct sum operation
  • Elements of $K(X)$ are formal differences of vector bundles, with $[E] - [F]$ representing the stable equivalence class of $E \oplus F^\perp$
• The reduced K-theory $\tilde{K}(X)$ is defined as the kernel of the map $K(X) \to K(*)$ induced by the map $X \to *$ to a point
  • $\tilde{K}(X)$ captures the essential information about vector bundles that vanish on the base point
• The K-theory functor satisfies the homotopy invariance property: if $f, g: X \to Y$ are homotopic maps, then the induced maps $f^*, g^*: K(Y) \to K(X)$ are equal
• The K-theory functor is also a generalized cohomology theory, satisfying the Eilenberg-Steenrod axioms except for the dimension axiom
  • This allows for the use of powerful tools from algebraic topology in the study of K-theory
• The Chern character provides a connection between K-theory and ordinary cohomology, mapping $K(X)$ to the even degree cohomology of $X$ with rational coefficients
• The Thom isomorphism theorem relates the K-theory of a vector bundle to the K-theory of its base space, providing a key tool in computations

The Bott Periodicity Theorem

• Bott periodicity establishes a periodic relationship between the K-groups of a space and its suspensions
• For a compact Hausdorff space $X$, there are natural isomorphisms:
  • $K(X) \cong K(S^2X)$ (complex Bott periodicity)
  • $KO(X) \cong KO(S^8X)$ (real Bott periodicity)
• These isomorphisms are given by the Bott maps, which are constructed using the Clifford algebra bundles
• The periodicity theorem implies that the K-groups of a space can be computed from the K-groups of its low-dimensional suspensions
  • For example, $K(S^n) \cong \mathbb{Z}$ for even $n$ and $K(S^n) \cong 0$ for odd $n$
• Bott periodicity can be interpreted as a manifestation of the periodicity of Clifford algebras
  • The Clifford algebra $Cl_n$ is isomorphic to a matrix algebra over $\mathbb{R}$, $\mathbb{C}$, or $\mathbb{H}$, depending on the value of $n$ modulo 8
• The periodicity theorem has far-reaching consequences in topology and geometry, simplifying computations and providing a unified perspective on various phenomena
• Bott periodicity has been generalized to other settings, such as equivariant K-theory and K-theory for operator algebras

Proof Techniques and Strategies

• The original proof of Bott periodicity by Raoul Bott used Morse theory and the study of geodesics on symmetric spaces
• Atiyah and Hirzebruch provided a more algebraic proof using the Thom isomorphism theorem and the properties of Clifford algebras
  • They constructed Clifford algebra bundles over spheres and used them to define the Bott maps
• The Atiyah-Bott-Shapiro construction relates Clifford modules to K-theory, providing a key tool in the proof of Bott periodicity
  • It establishes an isomorphism between the K-theory of a Clifford algebra bundle and the K-theory of its base space
• The use of the Thom isomorphism theorem allows for the reduction of the periodicity statement to a computation in the K-theory of Clifford algebras
• The proof relies on the periodicity of Clifford algebras, which is established using the structure theory of Clifford algebras and their representations
• Spectral sequences, such as the Atiyah-Hirzebruch spectral sequence, can be used to compute K-groups and study the relationship between K-theory and other cohomology theories
• The use of operator algebras and functional analysis techniques, such as the study of Fredholm operators and the index theorem, provides another perspective on the proof of Bott periodicity
• More recent proofs of Bott periodicity, such as the proof by Suslin and Voevodsky using motivic cohomology, offer new insights and connections to other areas of mathematics

Applications and Examples

• Bott periodicity has numerous applications in topology, geometry, and mathematical physics
• In topology, Bott periodicity is used to compute the K-groups of various spaces, such as spheres, projective spaces, and Grassmannians
  • For example, $K(\mathbb{C}P^n) \cong \mathbb{Z}^{n+1}$, with generators given by the powers of the tautological line bundle
• The Atiyah-Singer index theorem, which relates the index of an elliptic operator to topological invariants, relies on the machinery of K-theory and Bott periodicity
  • The index theorem has applications in differential geometry, topology, and mathematical physics
• In geometry, Bott periodicity is used to study the K-theory of flag varieties and homogeneous spaces
  • It provides a powerful tool for computing characteristic classes and understanding the topology of these spaces
• K-theory and Bott periodicity have applications in mathematical physics, particularly in the study of D-branes and string theory
  • The K-theory classification of D-branes and the study of Ramond-Ramond charges rely on the machinery of K-theory and Bott periodicity
• In noncommutative geometry, Bott periodicity is used to study the K-theory of C*-algebras and to classify noncommutative spaces
  • The Connes-Kasparov conjecture, which relates the K-theory of a C*-algebra to its primitive ideal space, is a generalization of Bott periodicity
• Bott periodicity has also found applications in algebraic topology, homotopy theory, and stable homotopy theory
  • It provides a powerful tool for computing stable homotopy groups and understanding the structure of the stable homotopy category

Connections to Other Mathematical Fields

• Topological K-theory and Bott periodicity have deep connections to various other areas of mathematics
• In algebraic topology, K-theory is related to other generalized cohomology theories, such as ordinary cohomology, bordism theory, and stable homotopy theory
  • The Chern character provides a link between K-theory and ordinary cohomology, while the Conner-Floyd isomorphism relates K-theory to bordism theory
• K-theory is closely tied to the theory of characteristic classes, which studies the obstruction to the existence of certain geometric structures on manifolds
  • The Chern classes, Stiefel-Whitney classes, and Pontryagin classes are all related to K-theory and can be studied using the machinery of Bott periodicity
• In algebraic geometry, K-theory is related to the study of algebraic cycles and motivic cohomology
  • The Grothendieck group of coherent sheaves on a scheme is an algebraic analog of topological K-theory, and the Bloch-Quillen formula relates it to Chow groups
• K-theory has important applications in operator algebras and noncommutative geometry
  • The K-theory of C*-algebras is used to classify noncommutative spaces, and the Baum-Connes conjecture relates it to the topology of the underlying space
• In mathematical physics, K-theory and Bott periodicity appear in the study of D-branes, string theory, and quantum field theory
  • The K-theory classification of D-brane charges and the study of Ramond-Ramond fields rely on the machinery of K-theory and Bott periodicity
• Bott periodicity has also been generalized to other settings, such as equivariant K-theory, twisted K-theory, and K-theory for operator algebras
  • These generalizations provide new insights and connections to other areas of mathematics, such as representation theory, index theory, and noncommutative geometry

Advanced Topics and Current Research

• Topological K-theory and Bott periodicity continue to be active areas of research, with many exciting developments and open problems
• Equivariant K-theory studies the K-theory of spaces with group actions, taking into account the symmetries of the space
  • Equivariant Bott periodicity has been established for compact Lie groups, and the Atiyah-Segal completion theorem relates equivariant K-theory to the representation ring of the group
• Twisted K-theory is a generalization of K-theory that takes into account the twisting of vector bundles by cohomology classes
  • It has applications in string theory and the study of D-branes, and is related to the theory of gerbes and differential cohomology
• The Baum-Connes conjecture, which relates the K-theory of a group C*-algebra to the topology of the classifying space of the group, is a major open problem in noncommutative geometry
  • It has been proven for a large class of groups, but the general case remains open
• The Connes-Kasparov conjecture, which relates the K-theory of a C*-algebra to its primitive ideal space, is another important open problem in noncommutative geometry
  • It is a generalization of Bott periodicity and has deep connections to the Baum-Connes conjecture and the Novikov conjecture
• The study of K-theory and Bott periodicity in the context of derived algebraic geometry and higher category theory has led to new insights and generalizations
  • The theory of spectral algebraic geometry and the study of K-theory in the context of ∞-categories provide new perspectives on these classical results
• The relationship between K-theory, Bott periodicity, and other areas of mathematics, such as homotopy theory, algebraic geometry, and mathematical physics, continues to be a fruitful area of research
  • New connections and applications are being discovered, and the interplay between these fields is leading to exciting new developments in mathematics