🪡K-Theory Unit 10 – K–Theory in Geometry and Topology

Foundations of K-Theory

• K-theory studies vector bundles over topological spaces and their associated invariants
• Originated from the work of Alexander Grothendieck in the 1950s on algebraic geometry
  • Grothendieck introduced the concept of K-groups to classify vector bundles
• Michael Atiyah and Friedrich Hirzebruch developed topological K-theory in the 1960s
  • Extended Grothendieck's ideas to the realm of topology
• K-theory provides a powerful tool for understanding the structure of vector bundles
• Connects various branches of mathematics, including topology, algebra, and analysis
• Plays a crucial role in the study of characteristic classes and index theory
• Has applications in physics, such as in the classification of topological insulators and superconductors

Topological K-Theory Basics

• Topological K-theory associates a graded ring, called the K-theory ring, to every compact Hausdorff space
• The elements of the K-theory ring represent stable equivalence classes of vector bundles over the space
• The K-theory ring is constructed using the Grothendieck completion process
  • Involves formally adding inverses to the monoid of isomorphism classes of vector bundles
• The rank of a vector bundle induces a ring homomorphism from the K-theory ring to the integers
• The reduced K-theory is defined for pointed spaces and is related to the unreduced K-theory via a split exact sequence
• The K-theory functor is contravariant, meaning it reverses the direction of continuous maps between spaces
• K-theory satisfies the homotopy invariance property, making it a powerful tool in homotopy theory

Vector Bundles and Their Role

• Vector bundles are the central objects of study in K-theory
• A vector bundle is a continuous family of vector spaces parametrized by a topological space (base space)
  • Each point in the base space is associated with a vector space (fiber)
• The trivial vector bundle is a product of the base space and a fixed vector space
• Isomorphism classes of vector bundles over a space form a monoid under the direct sum operation
• The tensor product of vector bundles induces a ring structure on the K-theory
• The pullback operation allows the transfer of vector bundles along continuous maps
• Vector bundles have numerous applications in geometry and physics
  • Tangent bundles, normal bundles, and cotangent bundles are examples of vector bundles in differential geometry

K-Groups and Operations

• The K-group of a compact Hausdorff space $X$, denoted by $K(X)$, is the Grothendieck group of the monoid of isomorphism classes of vector bundles over $X$
• The reduced K-group, denoted by $\tilde{K}(X)$, is defined for pointed spaces and is related to $K(X)$ via a split exact sequence
• The higher K-groups, $K^{-n}(X)$, are defined using the suspension operation on spaces
• The direct sum of vector bundles induces the addition operation in the K-group
• The tensor product of vector bundles induces the multiplication operation in the K-group, making it a ring
• The external tensor product allows the construction of vector bundles over product spaces
• The pullback operation on vector bundles induces a contravariant functoriality of K-groups
• The pushforward operation, related to the Thom isomorphism, allows the transfer of K-theory classes along certain maps

Bott Periodicity Theorem

• The Bott periodicity theorem is a fundamental result in K-theory, discovered by Raoul Bott in the 1950s
• It states that the K-groups of a space $X$ are periodic with period 2, i.e., $K^{-n}(X) \cong K^{-n-2}(X)$ for all $n$
• The theorem implies that the K-theory of a space is completely determined by its two initial K-groups, $K(X)$ and $K^{-1}(X)$
• Bott periodicity can be proven using the Atiyah-Bott-Shapiro construction, which involves Clifford algebras and Clifford modules
• The theorem has important consequences in topology and geometry
  • Allows the computation of K-groups of various spaces, such as spheres and projective spaces
• Bott periodicity is related to the periodicity of Clifford algebras and the eight-fold periodicity of real Clifford algebras
• The theorem has been generalized to other contexts, such as equivariant K-theory and twisted K-theory

Applications in Geometry

• K-theory has numerous applications in geometry, particularly in the study of manifolds and their invariants
• The Chern character provides a ring homomorphism from K-theory to rational cohomology, relating the two theories
• The Atiyah-Singer index theorem expresses the index of an elliptic operator on a manifold in terms of characteristic classes in K-theory
  • Has applications in the study of differential operators and the geometry of manifolds
• The Riemann-Roch theorem for algebraic curves can be generalized to higher dimensions using K-theory
• K-theory is used in the classification of vector bundles over manifolds and the computation of their characteristic classes
• The K-theoretic approach to the Novikov conjecture relates the higher signatures of manifolds to their K-theory
• K-theory plays a role in the study of positive scalar curvature metrics on manifolds and the Gromov-Lawson-Rosenberg conjecture

Connections to Other Mathematical Fields

• K-theory has deep connections to various branches of mathematics, including algebraic geometry, algebraic topology, and operator algebras
• In algebraic geometry, algebraic K-theory is used to study vector bundles over algebraic varieties and schemes
  • Related to the study of motives and the Grothendieck group of coherent sheaves
• In algebraic topology, K-theory is related to stable homotopy theory and the study of generalized cohomology theories
  • The Atiyah-Hirzebruch spectral sequence relates K-theory to ordinary cohomology
• K-theory of operator algebras, such as C*-algebras and von Neumann algebras, is a powerful tool in noncommutative geometry
  • Used in the classification of factors and the study of index theory for noncommutative spaces
• K-theory has applications in number theory, such as the study of algebraic cycles and the Birch-Swinnerton-Dyer conjecture
• The K-theory of group rings and the Baum-Connes conjecture relate K-theory to geometric group theory and the study of proper actions

Advanced Topics and Current Research

• Equivariant K-theory studies vector bundles over spaces with group actions, taking into account the symmetries of the space
  • Has applications in representation theory and the study of orbifolds
• Twisted K-theory incorporates the notion of twisting by a cohomology class, leading to new invariants and a generalization of the Atiyah-Singer index theorem
• The K-theory of categories, such as the Waldhausen K-theory of spaces and the Quillen K-theory of exact categories, provides a unified framework for studying K-theory in various contexts
• The relationship between K-theory and cyclic homology, as explored by Alain Connes and others, has led to important developments in noncommutative geometry
• The study of K-theory with coefficients, such as K-theory with finite coefficients or K-theory with local coefficients, has applications in topology and geometry
• The K-theory of structured ring spectra, such as the K-theory of E_∞ ring spectra, is an active area of research in homotopy theory
• The trace methods in K-theory, such as the Dennis trace and the cyclotomic trace, provide a connection between K-theory and topological cyclic homology
• Current research in K-theory includes the study of higher categorical structures, such as K-theory of ∞-categories and the K-theory of derived categories