5.3 Relationships between K-Theory and cohomology

The Chern character bridges K-theory and cohomology, providing a powerful tool for understanding complex vector bundles. It's a ring homomorphism that maps K-theory elements to rational cohomology, preserving key properties like tensor products and direct sums.

This connection allows us to use cohomological methods to compute K-theory groups, simplifying many calculations. However, it's important to note that this isomorphism is rational, meaning some torsion elements in K-theory may not be captured by rational cohomology.

Chern Character: K-theory and Cohomology

Definition and Properties

• The Chern character is a ring homomorphism from the K-theory of a topological space X to the rational cohomology of X, denoted as ch: K(X) → H^*(X;Q)
• For a complex vector bundle E over X, the Chern character is defined as ch(E) = rank(E) + c_1(E) + (1/2)c_1(E)^2 - c_2(E) + ..., where c_i(E) are the Chern classes of E
  • Example: For a line bundle L, ch(L) = 1 + c_1(L)
  • Example: For a rank 2 bundle E, ch(E) = 2 + c_1(E) + (1/2)c_1(E)^2 - c_2(E)
• The Chern character is a natural transformation between the K-theory functor and the rational cohomology functor, meaning it commutes with continuous maps between topological spaces
  • If f: X → Y is a continuous map, then ch(f^(E)) = f^(ch(E)) for any vector bundle E over Y

Compatibility with Vector Bundle Operations

• The Chern character is compatible with the tensor product of vector bundles, i.e., ch(E ⊗ F) = ch(E) ∧ ch(F), where ∧ is the cup product in cohomology
  • Example: For line bundles L and M, ch(L ⊗ M) = (1 + c_1(L)) ∧ (1 + c_1(M)) = 1 + c_1(L) + c_1(M) + c_1(L) ∧ c_1(M)
• The Chern character is also compatible with the direct sum of vector bundles, i.e., ch(E ⊕ F) = ch(E) + ch(F)
  • Example: For vector bundles E and F, ch(E ⊕ F) = (rank(E) + c_1(E) + ...) + (rank(F) + c_1(F) + ...) = (rank(E) + rank(F)) + (c_1(E) + c_1(F)) + ...
• These properties allow for the computation of the Chern character of complex vector bundles built from simpler ones using standard operations

K-theory vs Cohomology: Rational Isomorphism

Chern Character Induces Rational Isomorphism

• The Chern character induces an isomorphism between the K-theory of a space X tensored with the rational numbers and the rational cohomology of X, i.e., K(X) ⊗ Q ≅ H^*(X;Q)
  • This means that after tensoring with Q, the K-theory and cohomology groups become isomorphic vector spaces over Q
  • The isomorphism is given by the Chern character, which maps K(X) ⊗ Q to H^*(X;Q)
• This isomorphism holds for both complex and real K-theory, with the real case requiring a slight modification of the Chern character
  • For real K-theory, the Chern character is defined using the Pontryagin classes instead of the Chern classes

Consequences and Limitations

• The rational isomorphism allows for the computation of K-theory groups using cohomological methods, which are often easier to work with than direct K-theoretic calculations
  • Example: The K-theory of the complex projective space CP^n can be computed using the cohomology ring H^*(CP^n;Q) and the Chern character
• The isomorphism is not integral, meaning that there may be torsion elements in the K-theory that are not captured by the rational cohomology
  • Torsion elements are elements of finite order in the K-theory groups, which become zero when tensored with Q
  • Example: The K-theory of the real projective plane RP^2 has a torsion element of order 2, which is not detected by the rational cohomology

Atiyah-Hirzebruch Spectral Sequence and Chern Classes

Overview of the AHSS

• The Atiyah-Hirzebruch spectral sequence (AHSS) is a tool for computing the K-theory of a space X using its cellular filtration
  • The cellular filtration of X is a sequence of subspaces X_0 ⊂ X_1 ⊂ ... ⊂ X_n = X, where X_i is the i-skeleton of X
• The E_2 page of the AHSS is given by E_2^{p,q} = H^p(X; K^q(point)), where K^q(point) is the K-theory of a point, which is Z for even q and 0 for odd q
  • The E_2 page is a bigraded module over the cohomology ring of X
• The AHSS converges to the associated graded of the K-theory of X, with the filtration given by the skeletal filtration of X
  • The associated graded of a filtered group G is the direct sum of the quotients G_i/G_{i-1}, where G_i is the i-th filtration subgroup

Relation to Chern Classes

• The differentials in the AHSS are related to the Chern classes of the complex vector bundles over X
• The first differential d_2 is given by the first Chern class c_1
  • d_2: E_2^{p,q} → E_2^{p+2,q-1} is given by d_2(x) = c_1(x) for x ∈ E_2^{p,q} = H^p(X; K^q(point))
• The higher differentials are related to the higher Chern classes and the Steenrod operations in cohomology
  • The Steenrod operations are cohomology operations that generalize the cup product and the Bockstein homomorphism
  • Example: The second differential d_3 is related to the second Chern class c_2 and the Steenrod square Sq^2
• The AHSS provides a way to understand the K-theory of X in terms of its cohomology and characteristic classes, revealing the intricate interplay between these invariants

K-theory and Cohomology: Applications

Topological and Geometric Applications

• The Chern character and the rational isomorphism between K-theory and cohomology have been used to prove important results in topology and geometry
  • The Hopf invariant one problem, which characterizes the spheres that admit a division algebra structure, was solved using K-theory and the Adams operations
  • The Adams conjecture, which relates the Adams operations in K-theory to the Chern character, was proved using the AHSS and the Steenrod operations
• In algebraic geometry, the Grothendieck-Riemann-Roch theorem relates the Chern character of a coherent sheaf to its pushforward under a proper morphism, providing a powerful tool for computing Euler characteristics
  • The theorem generalizes the classical Riemann-Roch theorem for curves and surfaces to higher-dimensional varieties
  • It has applications in the study of moduli spaces of curves and the computation of intersection numbers

Applications in Mathematical Physics

• In index theory, the Atiyah-Singer index theorem expresses the index of an elliptic operator on a compact manifold in terms of characteristic classes, which can be computed using the Chern character and cohomology
  • The index theorem has applications in the study of differential equations, spectral theory, and quantum field theory
  • It relates analytical invariants (indices of operators) to topological invariants (characteristic classes)
• The K-theory of C*-algebras and its relation to cyclic cohomology have found applications in noncommutative geometry and the study of leaf spaces of foliations
  • Noncommutative geometry generalizes classical geometric concepts to noncommutative spaces, such as quantum spaces and leaf spaces of foliations
  • Cyclic cohomology is a cohomology theory for noncommutative algebras that generalizes the de Rham cohomology of manifolds
• In string theory, the K-theory of spacetime is related to the classification of D-brane charges, with the Chern character playing a role in the description of Ramond-Ramond fields
  • D-branes are extended objects in string theory that carry charges classified by K-theory
  • Ramond-Ramond fields are differential form fields in string theory that couple to D-branes and are described by the Chern character of K-theory classes