8.3 Applications to the computation of K-groups

The Conner-Floyd Chern character and Adams operations are powerful tools for computing K-groups. They bridge K-theory and cohomology, allowing us to calculate K-groups using cohomological data and detect torsion elements.

This section dives into practical applications, showing how to use these tools to compute K-groups for various spaces. We'll see examples of calculations for spheres, projective spaces, and more complex structures, connecting abstract theory to concrete results.

K-groups using Conner-Floyd Chern

Conner-Floyd Chern Character and Adams Operations

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• Conner-Floyd Chern character homomorphism maps K-theory to cohomology enables K-group computation using cohomological data
• Adams operations provide stable cohomology operations on K-theory offering additional structure and computational tools
  • Family of operations $\psi^k$ for each positive integer k
  • Satisfy properties like $\psi^k(x + y) = \psi^k(x) + \psi^k(y)$ and $\psi^k(xy) = \psi^k(x)\psi^k(y)$
• Combining Conner-Floyd Chern character and Adams operations allows K-group computation for spaces with known cohomology
  • Example: For a sphere $S^n$, use Chern character to map K-groups to cohomology, then apply Adams operations to determine torsion

Computational Techniques

• Atiyah-Hirzebruch spectral sequence used with Conner-Floyd Chern character computes K-groups of finite CW complexes
  • $E_2^{p,q} = H^p(X; \pi_q(K))$ converges to $K_{p+q}(X)$
  • Example: Computing $K^*(CP^n)$ using the spectral sequence
• Exact sequences facilitate K-group computation
  • Mayer-Vietoris sequence splits space into simpler pieces
    • For $X = A \cup B$, sequence: $... \to K^n(X) \to K^n(A) \oplus K^n(B) \to K^n(A \cap B) \to K^{n+1}(X) \to ...$
  • Long exact sequence of a pair relates K-groups of space and subspace
    • For pair $(X,A)$, sequence: $... \to K^n(X,A) \to K^n(X) \to K^n(A) \to K^{n+1}(X,A) \to ...$
• Torsion detection and computation in K-groups uses Adams operations and eigenvalues
  • Example: $\psi^k$ acts as multiplication by $k^i$ on $\tilde{K}^0(S^{2i})$, helping identify torsion elements

Equivariant K-theory Computations

• Equivariant K-group computation requires additional techniques
  • Representation theory utilized to analyze group actions on vector bundles
  • Fixed point formulas (Lefschetz fixed point theorem) apply to equivariant settings
    • Example: For a finite group G acting on a space X, $K_G^*(X) \cong K^*(X/G)$ if the action is free
  • Character formulas and localization techniques aid in explicit calculations
    • Example: Computing $K_G^*(pt)$ for a compact Lie group G using its representation ring

Interpretation of K-groups

Vector Bundle Classifications

• K-groups provide information about stable isomorphism classes of vector bundles over a space
• $K^0(X)$ rank corresponds to number of distinct stable isomorphism classes of vector bundles over X
  • Example: For a point, $K^0(pt) \cong \mathbb{Z}$ represents the stable isomorphism class of trivial bundles
• Torsion elements in $K^0(X)$ represent vector bundles becoming trivial after taking direct sums with themselves a certain number of times
  • Example: Hopf line bundle over $CP^1$ generates torsion element in $\tilde{K}^0(CP^1)$

Higher K-groups and Geometric Interpretations

• $K^1(X)$ interpreted in terms of automorphisms of trivial bundles or clutching functions for vector bundles over suspended spaces
  • Example: $K^1(S^1) \cong \mathbb{Z}$ corresponds to winding number of maps $S^1 \to GL_n(\mathbb{C})$
• Bott periodicity theorem relates $K^0$ and $K^1$, allowing interpretation of higher K-groups in terms of vector bundles
  • $K^n(X) \cong K^{n+2}(X)$ for all n
• Thom isomorphism in K-theory relates K-groups of a space to those of its Thom space, providing geometric interpretations
  • For a vector bundle E over X, $K^*(X) \cong K^*(Th(E))$, where Th(E) denotes the Thom space
• Geometric realizations of K-theory classes used to interpret computational results
  • Projective modules over C(X) correspond to vector bundles over X
  • Families of Fredholm operators represent elements in K-theory
    • Example: Index bundle of a family of elliptic operators on a manifold

K-group computation methods

Spectral Sequence Approaches

• Atiyah-Hirzebruch spectral sequence provides systematic approach to computing K-groups using cohomological information
  • Requires complex calculations for higher differentials
  • Example: Computing $K^*(CP^\infty)$ using the spectral sequence and its collapse at the $E_2$ page
• Conner-Floyd Chern character method effective for rational computations and spaces with torsion-free cohomology
  • May not capture all torsion information
  • Example: Using Chern character to compute $K^*(S^n) \otimes \mathbb{Q}$

Geometric and Analytical Methods

• Index theory and Atiyah-Singer index theorem offer powerful tools for computing K-groups of manifolds
  • Particularly effective in the presence of additional geometric structures
  • Example: Computing K-theory class of Dirac operator on a spin manifold
• Representation theory techniques essential for computing equivariant K-groups
  • Not applicable to non-equivariant settings
  • Example: Using character formulas to compute $K_G^*(G/H)$ for compact Lie groups G and H

Algebraic and Computational Techniques

• Algebraic methods provide general frameworks for computation
  • Exact sequences (Mayer-Vietoris, long exact sequence of a pair)
  • Spectral sequences (Atiyah-Hirzebruch, Adams spectral sequence)
  • Require specific geometric or topological input
  • Example: Using Mayer-Vietoris sequence to compute $K^*(S^n)$
• Computational techniques based on Adams operations effective for detecting and computing torsion in K-groups
  • Limited by complexity of operations
  • Example: Using Adams operations to determine torsion in $K^*(RP^n)$

Applications of K-groups

Topological Applications

• K-theory obstruction theory determines existence and classification of vector bundles over given space
  • Example: Using K-theory to classify complex line bundles over spheres
• K-group computations provide information about stable homotopy groups of spheres through J-homomorphism and Adams' e-invariant
  • J-homomorphism: $J: \pi_i(O) \to \pi_i^s(S^0)$
  • Example: Computing $\pi_4^s(S^0)$ using K-theory and J-homomorphism
• K-theory computations applied to study immersions and embeddings of manifolds using Atiyah-Hirzebruch obstruction theory
  • Example: Determining the minimal dimension for immersing $RP^n$ in Euclidean space

Geometric and Analytical Applications

• Index of elliptic operators on manifolds computed using K-theory leads to applications in differential geometry and global analysis
  • Example: Computing index of Dirac operator on a spin manifold
• K-theory essential in formulation and proof of Atiyah-Singer index theorem relating analytical and topological invariants of manifolds
  • $\text{ind}(D) = \int_M \text{ch}(\sigma(D)) \text{Td}(TM)$
• Equivariant K-group computation applied to study group actions on manifolds and derive fixed point theorems
  • Example: Using equivariant K-theory to prove the Lefschetz fixed point theorem

Noncommutative Geometry and C*-algebras

• K-theory computations play crucial role in classification of C*-algebras and study of noncommutative geometry
  • Bridge gap between topology and operator algebras
  • Example: Computing K-theory of irrational rotation algebras
• K-theory used to formulate and prove index theorems in noncommutative settings
  • Example: Connes' index theorem for foliations