5.2 The Chern character and its properties

The Chern character is a powerful tool in K-theory, bridging the gap between vector bundles and cohomology. It's a ring homomorphism that maps K-theory to rational cohomology, providing a way to classify vector bundles and compute K-theory rings.

This concept is crucial for understanding the relationship between algebraic and topological properties of vector bundles. The Chern character's properties, like multiplicativity and rational isomorphism, make it invaluable for calculations in K-theory and cohomology.

Chern character construction

Definition and formula

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• The Chern character is a ring homomorphism from the K-theory ring to the cohomology ring with rational coefficients, ch: K(X) → H^even(X;Q)
• Defined in terms of Chern classes for a complex vector bundle E, 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
• Natural transformation between the functors K(−) and H^even(−;Q), compatible with pullbacks and the ring structures

Extension to higher K-theory

• The Chern character extends to higher K-theory groups using the Atiyah-Hirzebruch spectral sequence
• Allows for the computation of higher K-theory groups in terms of cohomology groups
• Provides a connection between the algebraic and topological properties of vector bundles

Chern character properties

Ring homomorphism and multiplicativity

• The Chern character is a ring homomorphism, ch(E ⊕ F) = ch(E) + ch(F) and ch(E ⊗ F) = ch(E) ∪ ch(F), where ⊕ is the direct sum, ⊗ is the tensor product, and ∪ is the cup product in cohomology
• Multiplicative property, ch(E ⊗ F) = ch(E) ∪ ch(F), follows from the Whitney product formula for Chern classes
• Examples: ch(E ⊕ F) = ch(E) + ch(F) for vector bundles E and F over a manifold X, ch(L ⊗ L') = ch(L) ∪ ch(L') for line bundles L and L' over a complex projective space

Rational isomorphism and Grothendieck-Riemann-Roch theorem

• The Chern character is a rational isomorphism, ch: K(X) ⊗ Q ≅ H^even(X;Q), becoming an isomorphism after tensoring with the rational numbers
• Satisfies the Grothendieck-Riemann-Roch theorem, relating the pushforward in K-theory to the pushforward in cohomology via the Chern character and the Todd class
• Example: For a proper morphism f: X → Y, the Grothendieck-Riemann-Roch theorem states that ch(f_!(E)) = f_(ch(E) ∪ td(TX)) for a vector bundle E over X, where f_! is the pushforward in K-theory, f_ is the pushforward in cohomology, and td(TX) is the Todd class of the tangent bundle of X

Chern character and vector bundles

Classification of vector bundles

• The Chern character provides a complete invariant for complex vector bundles up to stable equivalence, two vector bundles are stably equivalent if and only if they have the same Chern character
• Rational K-theory of a space X, K(X) ⊗ Q, is isomorphic to the rational cohomology of X in even degrees, H^even(X;Q), via the Chern character
• Chern character classifies vector bundles over certain spaces (spheres and projective spaces) by computing their K-theory rings and the Chern characters of their generators

Examples of vector bundle classification

• Complex projective space CP^n: The tautological line bundle L over CP^n has ch(L) = 1 + h, where h is the generator of H^2(CP^n;Z), and K(CP^n) ≅ Z[L]/(L^(n+1)-1)
• Complex Grassmannian Gr(k,n): The tautological vector bundle S over Gr(k,n) has ch(S) = k + c_1(S) + (1/2)(c_1(S)^2 - 2c_2(S)) + ..., and the K-theory ring of Gr(k,n) is generated by the exterior powers of S

Chern character in K-theory

Computing K-theory rings

• The Chern character computes the K-theory rings of various spaces (spheres, projective spaces, and Grassmannians) by relating them to their cohomology rings
• Complex projective space CP^n: K(CP^n) ≅ Z[x]/(x^(n+1)), where x is the class of the tautological line bundle, and ch(x) = 1 + h, where h is the generator of H^2(CP^n;Z)
• Complex Grassmannian Gr(k,n): The K-theory ring is generated by the classes of the tautological vector bundles, and the Chern character expresses their relations in terms of the cohomology ring

Bott periodicity theorem

• The Chern character proves the Bott periodicity theorem, stating that the K-theory of a point is periodic with period 2, K(point) ≅ K(S^2) ≅ K(S^4) ≅ ... ≅ Z[u, u^(-1)], where u is the Bott generator
• Bott periodicity connects the K-theory of different dimensional spaces and provides a powerful tool for computing K-theory rings
• Example: The Bott generator u ∈ K(S^2) has ch(u) = 1 + h, where h is the generator of H^2(S^2;Z), and the Bott periodicity map β: K(X) → K(X × S^2) is given by β(E) = E ⊗ u for a vector bundle E over X