10.2 Applications to vector bundle classification

K-Theory provides powerful tools for classifying vector bundles over topological spaces. It introduces the group K(X), which encapsulates the structure of vector bundles over X, and explores concepts like stable equivalence and Chern classes.

The periodicity theorem for complex K-Theory reveals a surprising connection between K(X) and K(Σ^2X). This result, along with real K-Theory and its applications, offers deep insights into the classification and properties of vector bundles.

Classifying vector bundles with K-Theory

The K-Theory group K(X) and stable equivalence

• The group K(X) is defined as the Grothendieck group of the monoid of isomorphism classes of vector bundles over X under the direct sum operation
• Two vector bundles E and F over X are stably equivalent if there exist trivial bundles ε^n and ε^m such that E ⊕ ε^n ≅ F ⊕ ε^m
  • The set of stable equivalence classes of vector bundles over X forms an abelian group under the direct sum operation, denoted as K^0(X)
• The reduced K-Theory group, denoted as K̃(X), is defined as the kernel of the rank homomorphism rk: K(X) → Z, where rk(E) is the dimension of the fiber of the vector bundle E

Chern classes and the Chern character

• The first Chern class c_1(E) of a complex line bundle E over X is an element of the second cohomology group H^2(X; Z)
  • The first Chern class induces a group homomorphism c_1: K(X) → H^2(X; Z) for X a CW complex
• The Chern character is a ring homomorphism ch: K(X) → H^*(X; Q) that maps the K-Theory group to the rational cohomology ring of X
  • It is given by ch(E) = rk(E) + c_1(E) + (1/2)c_1(E)^2 + ... for a complex vector bundle E
  • The Chern character is an isomorphism after tensoring with the rational numbers Q, i.e., ch: K(X) ⊗ Q → H^*(X; Q) is an isomorphism of rings

Periodicity theorem for complex K-Theory

Statement and proof of the periodicity theorem

• The periodicity theorem states that for any compact Hausdorff space X, there is a natural isomorphism K(X) ≅ K(Σ^2X), where Σ^2X is the double suspension of X
• The proof of the periodicity theorem involves the following key steps:
  • Construct a natural transformation α: K(X) → K(Σ^2X) using the clutching construction and the Bott generator
  • Show that α is an isomorphism for X = S^n, the n-dimensional sphere, using the Atiyah-Hirzebruch spectral sequence
  • Extend the result to all compact Hausdorff spaces X using the Mayer-Vietoris sequence and the five lemma

Key concepts in the proof

• The Bott generator is a specific element β ∈ K(S^2) that generates the reduced K-Theory group K̃(S^2) ≅ Z
• The clutching construction is a method for constructing vector bundles over the suspension ΣX of a space X by specifying a clutching function f: X → GL(n, C)
  • For example, the Hopf bundle over S^2 can be constructed using the clutching function f: S^1 → GL(1, C) given by f(z) = z
• The Atiyah-Hirzebruch spectral sequence relates the K-Theory of a space X to its cellular cohomology, providing a means to compute K(X) in terms of H^*(X; Z)

K-Theory for real vector bundles

Real K-Theory groups and Stiefel-Whitney classes

• The group KO(X) is defined as the Grothendieck group of the monoid of isomorphism classes of real vector bundles over X under the direct sum operation
• The reduced real K-Theory group, denoted as K̃O(X), is defined as the kernel of the rank homomorphism rk: KO(X) → Z
• The first Stiefel-Whitney class w_1(E) of a real line bundle E over X is an element of the first cohomology group H^1(X; Z/2Z)
  • The first Stiefel-Whitney class induces a group homomorphism w_1: KO(X) → H^1(X; Z/2Z) for X a CW complex

Real periodicity theorem and its implications

• The real periodicity theorem states that for any compact Hausdorff space X, there is a natural isomorphism KO(X) ≅ KO(Σ^8X), where Σ^8X is the 8-fold suspension of X
  • The real periodicity theorem implies that the groups KO(S^n) are periodic with period 8, i.e., KO(S^n) ≅ KO(S^(n+8)) for all n ≥ 0
• The group KO(X) contains information about the orientability and spin structures of real vector bundles over X
  • A real vector bundle E over X is orientable if and only if w_1(E) = 0
  • A real vector bundle E over X admits a spin structure if and only if w_1(E) = 0 and w_2(E) = 0, where w_2(E) is the second Stiefel-Whitney class of E

K-Theory applications for vector bundle structure

K-Theory of spheres and projective spaces

• The K-Theory of the n-dimensional sphere S^n is given by K(S^n) ≅ Z for n even and K(S^n) ≅ 0 for n odd
  • This implies that every complex vector bundle over S^n is stably trivial for n odd
• The K-Theory of the complex projective space CP^n is given by K(CP^n) ≅ Z[x]/(x^(n+1)), where x is the class of the tautological line bundle over CP^n
  • This ring structure allows for the computation of the K-Theory of CP^n-bundles
• The K-Theory of the real projective space RP^n is given by KO(RP^n) ≅ Z/2Z for n odd and KO(RP^n) ≅ Z ⊕ Z/2Z for n even
  • The generator of the Z/2Z factor corresponds to the canonical line bundle over RP^n

K-Theory of product spaces and the Atiyah-Hirzebruch spectral sequence

• The K-Theory of a product space X × Y is related to the K-Theory of the factors by the external tensor product: K(X × Y) ≅ K(X) ⊗ K(Y)
  • This allows for the computation of the K-Theory of product spaces (Cartesian products) using the K-Theory of the individual factors
• The Atiyah-Hirzebruch spectral sequence can be used to compute the K-Theory of a space X in terms of its cellular cohomology
  • For example, if X is a CW complex with only even-dimensional cells, then K(X) ≅ H^*(X; Z)
  • The Atiyah-Hirzebruch spectral sequence also provides a means to study the filtration of K(X) by the dimension of the cells in X, leading to the notion of the γ-filtration in K-Theory