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 ψ k \psi^k ψ k for each positive integer k
Satisfy properties like ψ k ( x + y ) = ψ k ( x ) + ψ k ( y ) \psi^k(x + y) = \psi^k(x) + \psi^k(y) ψ k ( x + y ) = ψ k ( x ) + ψ k ( y ) and ψ k ( x y ) = ψ k ( x ) ψ k ( y ) \psi^k(xy) = \psi^k(x)\psi^k(y) ψ k ( x y ) = ψ k ( x ) ψ 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 S^n 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 ; π q ( K ) ) E_2^{p,q} = H^p(X; \pi_q(K)) E 2 p , q = H p ( X ; π q ( K )) converges to K p + q ( X ) K_{p+q}(X) K p + q ( X )
Example: Computing K ∗ ( C P n ) K^*(CP^n) K ∗ ( C P n ) using the spectral sequence
Exact sequences facilitate K-group computation
Mayer-Vietoris sequence splits space into simpler pieces
For X = A ∪ B X = A \cup B X = A ∪ B , sequence: . . . → K n ( X ) → K n ( A ) ⊕ K n ( B ) → K n ( A ∩ B ) → K n + 1 ( X ) → . . . ... \to K^n(X) \to K^n(A) \oplus K^n(B) \to K^n(A \cap B) \to K^{n+1}(X) \to ... ... → K n ( X ) → K n ( A ) ⊕ K n ( B ) → K n ( A ∩ B ) → K n + 1 ( X ) → ...
Long exact sequence of a pair relates K-groups of space and subspace
For pair ( X , A ) (X,A) ( X , A ) , sequence: . . . → K n ( X , A ) → K n ( X ) → K n ( A ) → K n + 1 ( X , A ) → . . . ... \to K^n(X,A) \to K^n(X) \to K^n(A) \to K^{n+1}(X,A) \to ... ... → K n ( X , A ) → K n ( X ) → K n ( A ) → K n + 1 ( X , A ) → ...
Torsion detection and computation in K-groups uses Adams operations and eigenvalues
Example: ψ k \psi^k ψ k acts as multiplication by k i k^i k i on K ~ 0 ( S 2 i ) \tilde{K}^0(S^{2i}) K ~ 0 ( S 2 i ) , 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 ) ≅ K ∗ ( X / G ) K_G^*(X) \cong K^*(X/G) K G ∗ ( X ) ≅ K ∗ ( X / G ) if the action is free
Character formulas and localization techniques aid in explicit calculations
Example: Computing K G ∗ ( p t ) K_G^*(pt) 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 ) K^0(X) K 0 ( X ) rank corresponds to number of distinct stable isomorphism classes of vector bundles over X
Example: For a point, K 0 ( p t ) ≅ Z K^0(pt) \cong \mathbb{Z} K 0 ( pt ) ≅ Z represents the stable isomorphism class of trivial bundles
Torsion elements in K 0 ( X ) K^0(X) K 0 ( X ) represent vector bundles becoming trivial after taking direct sums with themselves a certain number of times
Example: Hopf line bundle over C P 1 CP^1 C P 1 generates torsion element in K ~ 0 ( C P 1 ) \tilde{K}^0(CP^1) K ~ 0 ( C P 1 )
Higher K-groups and Geometric Interpretations
K 1 ( X ) K^1(X) 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 ) ≅ Z K^1(S^1) \cong \mathbb{Z} K 1 ( S 1 ) ≅ Z corresponds to winding number of maps S 1 → G L n ( C ) S^1 \to GL_n(\mathbb{C}) S 1 → G L n ( C )
Bott periodicity theorem relates K 0 K^0 K 0 and K 1 K^1 K 1 , allowing interpretation of higher K-groups in terms of vector bundles
K n ( X ) ≅ K n + 2 ( X ) K^n(X) \cong K^{n+2}(X) K n ( X ) ≅ 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 ) ≅ K ∗ ( T h ( E ) ) K^*(X) \cong K^*(Th(E)) K ∗ ( X ) ≅ K ∗ ( T h ( 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 ∗ ( C P ∞ ) K^*(CP^\infty) K ∗ ( C P ∞ ) using the spectral sequence and its collapse at the E 2 E_2 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 ) ⊗ Q K^*(S^n) \otimes \mathbb{Q} K ∗ ( S n ) ⊗ 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 ) K_G^*(G/H) 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 ) K^*(S^n) 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 ∗ ( R P n ) K^*(RP^n) K ∗ ( R P 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 : π i ( O ) → π i s ( S 0 ) J: \pi_i(O) \to \pi_i^s(S^0) J : π i ( O ) → π i s ( S 0 )
Example: Computing π 4 s ( S 0 ) \pi_4^s(S^0) π 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 R P n RP^n R P 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
ind ( D ) = ∫ M ch ( σ ( D ) ) Td ( T M ) \text{ind}(D) = \int_M \text{ch}(\sigma(D)) \text{Td}(TM) ind ( D ) = ∫ M ch ( σ ( D )) 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