🪡K-Theory Unit 1 – Introduction to Vector Bundles and K–Theory

What's the Big Idea?

• K-Theory provides a powerful framework for studying vector bundles and their properties
• Combines ideas from algebraic topology, differential geometry, and abstract algebra to gain deeper insights into the structure of vector bundles
• Allows for the classification of vector bundles up to isomorphism using K-groups, which are abelian groups associated with a topological space
• Enables the computation of important invariants (characteristic classes) that capture essential information about the geometry and topology of vector bundles
• Provides a unified language for understanding and comparing vector bundles across different contexts (smooth manifolds, algebraic varieties, etc.)
  • Facilitates the transfer of ideas and techniques between these different settings
• Connects to other areas of mathematics, such as index theory and operator algebras, leading to fruitful interactions and applications
• Serves as a bridge between classical algebraic topology and modern developments in geometry and physics (gauge theory, string theory)

Key Concepts and Definitions

• Vector bundle: a family of vector spaces parametrized by a topological space, with a locally trivial structure
  • Consists of a total space $E$, a base space $B$, and a projection map $\pi: E \to B$
  • Each fiber $\pi^{-1}(b)$ is a vector space isomorphic to a fixed model space (typically $\mathbb{R}^n$ or $\mathbb{C}^n$)
• Sections of a vector bundle: continuous maps $s: B \to E$ such that $\pi \circ s = \text{id}_B$, assigning a vector to each point in the base space
• Trivial bundle: a vector bundle isomorphic to the product bundle $B \times \mathbb{R}^n$ (or $B \times \mathbb{C}^n$)
• Isomorphism of vector bundles: a fiber-preserving homeomorphism between the total spaces that restricts to linear isomorphisms on each fiber
• K-group: an abelian group associated with a topological space $X$, denoted $K(X)$, whose elements represent stable isomorphism classes of vector bundles over $X$
  • The group operation is induced by the direct sum of vector bundles
• Reduced K-theory: a variant of K-theory that assigns the trivial group to a point, denoted $\tilde{K}(X)$
• Characteristic classes: cohomology classes associated with vector bundles that capture geometric and topological properties (Chern classes, Pontryagin classes, Euler class)

Historical Context and Development

• Origins in the work of Alexander Grothendieck in the 1950s, who introduced K-theory for algebraic varieties as a tool for studying coherent sheaves
• Michael Atiyah and Friedrich Hirzebruch developed topological K-theory in the 1960s, extending Grothendieck's ideas to the context of topological spaces and vector bundles
  • Their seminal paper "Vector bundles and homogeneous spaces" (1961) laid the foundations for the subject
• Atiyah and Isadore Singer's work on the index theorem (1963) revealed deep connections between K-theory, differential geometry, and elliptic differential operators
• The Atiyah-Hirzebruch spectral sequence, introduced in their 1961 paper, provided a powerful computational tool for calculating K-groups
• Further developments by Raoul Bott, Graeme Segal, and others in the 1960s and 1970s expanded the scope and applications of K-theory
  • Bott periodicity, the Thom isomorphism, and the Segal conjecture are notable examples
• In the 1980s and beyond, K-theory found applications in diverse areas such as operator algebras (C*-algebras, von Neumann algebras), mathematical physics (gauge theory, string theory), and algebraic geometry (motivic cohomology)

Vector Bundles: The Basics

• A vector bundle $\pi: E \to B$ consists of a total space $E$, a base space $B$, and a projection map $\pi$ satisfying certain local triviality conditions
  • For each point $b \in B$, there exists a neighborhood $U$ and a homeomorphism $\phi: \pi^{-1}(U) \to U \times \mathbb{R}^n$ (or $U \times \mathbb{C}^n$) such that $\pi = \text{pr}_1 \circ \phi$
• The rank of a vector bundle is the dimension of its fibers (the vector spaces $\pi^{-1}(b)$)
• Examples of vector bundles include the tangent bundle and normal bundle of a smooth manifold, the tautological line bundle over projective space, and the Möbius band
• Operations on vector bundles:
  • Direct sum: given bundles $E_1$ and $E_2$ over $B$, their direct sum $E_1 \oplus E_2$ is a new bundle with fibers $(E_1 \oplus E_2)_b = (E_1)_b \oplus (E_2)_b$
  • Tensor product: the tensor product $E_1 \otimes E_2$ has fibers $(E_1 \otimes E_2)_b = (E_1)_b \otimes (E_2)_b$
  • Dual bundle: the dual bundle $E^*$ has fibers $(E^*)_b = ((E)_b)^*$, the dual vector space
• The Whitney sum formula relates the characteristic classes of a direct sum to those of its summands
• The pullback construction allows for the transfer of vector bundles along continuous maps between topological spaces

K-Theory: A New Perspective

• K-theory associates an abelian group $K(X)$ to each topological space $X$, whose elements represent stable isomorphism classes of vector bundles over $X$
  • Two vector bundles $E_1$ and $E_2$ are stably isomorphic if there exist trivial bundles $\varepsilon_1$ and $\varepsilon_2$ such that $E_1 \oplus \varepsilon_1 \cong E_2 \oplus \varepsilon_2$
• The group operation in $K(X)$ is induced by the direct sum of vector bundles, with the trivial bundle serving as the identity element
• Reduced K-theory $\tilde{K}(X)$ is defined as the kernel of the map $K(X) \to K(\text{point})$ induced by the constant map $X \to \text{point}$
  • Fits into a split short exact sequence $0 \to \tilde{K}(X) \to K(X) \to \mathbb{Z} \to 0$, with the rank function providing a splitting
• K-theory is a generalized cohomology theory, satisfying the Eilenberg-Steenrod axioms (except the dimension axiom)
  • Functorial with respect to continuous maps, with induced homomorphisms $f^*: K(Y) \to K(X)$ for $f: X \to Y$
  • Satisfies the homotopy invariance and Mayer-Vietoris sequence axioms
• The Bott periodicity theorem states that $K(X) \cong K(\Sigma^2 X)$, where $\Sigma^2 X$ is the double suspension of $X$
  • Implies that the K-groups of a space exhibit a periodic pattern, with $K^{-n}(X) \cong K^{-n-2}(X)$
• The Thom isomorphism theorem relates the K-theory of a vector bundle to that of its base space, providing a powerful computational tool

Important Theorems and Proofs

• The Atiyah-Hirzebruch spectral sequence relates the K-theory of a space to its ordinary cohomology, providing a means for computing K-groups
  • The $E_2$ page of the spectral sequence is given by $E_2^{p,q} = H^p(X; K^q(\text{point}))$, with differentials $d_r: E_r^{p,q} \to E_r^{p+r,q-r+1}$
  • Converges to the associated graded of a filtration on $K(X)$
• The Thom isomorphism theorem states that for a rank $n$ vector bundle $\pi: E \to B$, there is an isomorphism $\phi: K(B) \to \tilde{K}(E)$, called the Thom class
  • The Thom class is natural with respect to pullbacks and satisfies a multiplicative property with respect to the tensor product of bundles
• The Chern character is a ring homomorphism $\text{ch}: K(X) \to H^*(X; \mathbb{Q})$ that relates K-theory to rational cohomology
  • Defined in terms of the Chern classes of a vector bundle, with $\text{ch}(E) = \sum_{i=0}^{\infty} \frac{1}{i!}c_i(E)$
  • Becomes an isomorphism after tensoring with the rational numbers
• The Adams operations $\psi^k: K(X) \to K(X)$ are ring homomorphisms that provide a way to study the structure of K-theory
  • Defined in terms of the exterior powers of a vector bundle, with $\psi^k(E) = \sum_{i=0}^{\text{rank}(E)} (-1)^i \binom{k}{i} \Lambda^i(E)$
  • Satisfy the identity $\psi^k \circ \psi^l = \psi^{kl}$ and are related to the Chern character by $\text{ch} \circ \psi^k = k^* \circ \text{ch}$, where $k^*$ is the map induced by the multiplication by $k$ on rational cohomology
• The Segal conjecture, proved by Graeme Segal in the 1970s, provides a description of the K-theory of the classifying space $BG$ of a compact Lie group $G$ in terms of the representation ring $R(G)$

Real-World Applications

• K-theory has found applications in various branches of physics, particularly in the study of topological phases of matter and the classification of topological insulators and superconductors
  • The K-theory of the Brillouin zone (a torus in momentum space) captures the topological properties of electronic band structures
  • The bulk-boundary correspondence relates the K-theory of the bulk material to the existence of topologically protected edge states
• In string theory, K-theory provides a framework for understanding D-brane charges and the classification of Ramond-Ramond fields
  • The K-theory of spacetime is used to define the charge groups of D-branes, with stable isomorphism classes of vector bundles corresponding to different brane configurations
• K-theory has also been applied to the study of symmetry-protected topological phases and the classification of crystalline solids
  • The K-theory of the space of symmetry-preserving Hamiltonians captures the topological invariants that distinguish different phases
• In algebraic geometry, K-theory is used to study the category of coherent sheaves on a scheme or algebraic variety
  • The Grothendieck group of the category of coherent sheaves, denoted $K_0(X)$, provides a useful invariant for studying the geometry of the variety
  • Higher K-groups, such as $K_1(X)$ and $K_2(X)$, capture more refined information about the category of vector bundles and their automorphisms
• K-theory has also found applications in operator algebras and noncommutative geometry, where it is used to study the structure of C*-algebras and their modules
  • The K-theory of a C*-algebra $A$, denoted $K_0(A)$ and $K_1(A)$, provides a powerful invariant for distinguishing nonisomorphic algebras and understanding their properties

Common Pitfalls and How to Avoid Them

• Confusing stable isomorphism with isomorphism: remember that K-theory captures information up to stable equivalence, which allows for the addition of trivial bundles
  • To avoid this, always consider bundles up to stable isomorphism and be mindful of the role of trivial bundles in the equivalence relation
• Neglecting the role of the base space: the K-theory of a space depends crucially on the topology of the base, not just the fibers
  • Always consider the full data of a vector bundle, including the base space and the projection map, rather than just focusing on the fibers
• Misapplying the Bott periodicity theorem: the periodicity applies to the K-groups of a space, not to the space itself
  • Be careful when using Bott periodicity to compute K-groups, and make sure to apply the theorem correctly to the K-groups rather than the underlying topological space
• Overlooking the role of compact support: when working with non-compact spaces, it is often necessary to consider vector bundles with compact support or to use compactly supported K-theory
  • Be mindful of the support of the bundles you are working with, and use the appropriate version of K-theory for the situation at hand
• Mishandling the signs and gradings in the Mayer-Vietoris sequence: the connecting homomorphism in the Mayer-Vietoris sequence involves a sign that depends on the degree of the K-group
  • Pay close attention to the signs and gradings when using the Mayer-Vietoris sequence to compute K-groups, and be consistent in your choice of conventions
• Forgetting the hypotheses of theorems: many important results in K-theory, such as the Thom isomorphism and the Chern character, have specific hypotheses that must be satisfied for the theorems to hold
  • Always check that the hypotheses of a theorem are satisfied before applying it, and be aware of any limitations or restrictions on its use
• Confusing the different versions of K-theory: there are many variants of K-theory, such as topological, algebraic, and operator K-theory, each with its own specific definitions and properties
  • Be clear about which version of K-theory you are using, and make sure to use the appropriate definitions and results for that particular variant
• Not exploiting the full power of the tools: K-theory provides a rich toolkit for studying vector bundles and their properties, but it can be easy to overlook some of the more advanced techniques
  • Take the time to learn about the various tools and techniques available in K-theory, such as the Atiyah-Hirzebruch spectral sequence, the Adams operations, and the Chern character, and use them to their full potential in your work