10.1 K-Theory and characteristic classes

K-Theory connects vector bundles to cohomology, offering a powerful tool for studying geometric and topological spaces. Characteristic classes, like Chern classes, measure bundle twisting and provide a bridge between K-Theory and cohomology theories.

The Chern character, a ring homomorphism from K-Theory to rational cohomology, allows for cohomological computations in K-Theory. This connection, along with spectral sequences like Atiyah-Hirzebruch, helps uncover deep relationships between different mathematical structures.

Characteristic classes in K-Theory

Definition and role of characteristic classes

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• Characteristic classes are cohomology classes associated to vector bundles that measure the twisting or non-triviality of the bundle
• Provide a way to distinguish between different vector bundles over the same base space
• Allow for the construction of maps from K-Theory to various cohomology theories
• Examples of characteristic classes include Chern classes, Pontryagin classes, and Stiefel-Whitney classes

Types of characteristic classes

• Chern classes are associated to complex vector bundles
  • Live in the even degree cohomology of the base space with integer coefficients
  • Example: The first Chern class of a complex line bundle measures its degree of twisting
• Pontryagin classes are associated to real vector bundles
  • Live in the cohomology of the base space with integer coefficients in degrees divisible by 4
  • Example: The first Pontryagin class of a real vector bundle is related to its second Stiefel-Whitney class
• Stiefel-Whitney classes are associated to real vector bundles
  • Live in the cohomology of the base space with Z/2Z coefficients
  • Example: The first Stiefel-Whitney class of a real vector bundle determines its orientability

Chern character and K-Theory

Definition and properties of the Chern character

• The Chern character is a ring homomorphism from the K-Theory of a space X to the rational cohomology of X
• Defined using the Chern classes of complex vector bundles
• For a line bundle L, the Chern character is given by ch(L) = exp(c_1(L)), where c_1(L) is the first Chern class of L
• For a vector bundle E, 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

Relation to K-Theory

• The Chern character satisfies the properties of a ring homomorphism:
  • ch(E + F) = ch(E) + ch(F), where + denotes the sum of vector bundles
  • ch(E * F) = ch(E) * ch(F), where * denotes the tensor product of vector bundles
• Induces an isomorphism between the rational K-Theory of a space and its rational cohomology
• Allows for the computation of K-Theory groups using cohomological methods
• Example: The Chern character of a sum of line bundles is the sum of their exponentials of first Chern classes

K-Theory vs cohomology theories

K-Theory as a generalized cohomology theory

• K-Theory satisfies the Eilenberg-Steenrod axioms except for the dimension axiom
• Allows for the construction of K-Theory groups in negative degrees
• Example: The K-Theory of a point is Z in even degrees and 0 in odd degrees

Relation to ordinary cohomology

• The Chern character provides a link between K-Theory and ordinary cohomology theories
• Induces a rational isomorphism between K-Theory and rational cohomology
• The Atiyah-Hirzebruch spectral sequence relates the K-Theory of a space to its ordinary cohomology
• Example: The K-Theory of the complex projective space CP^n is isomorphic to a truncated polynomial ring over Z

Additional structures in K-Theory

• The Adams operations in K-Theory are analogous to the Steenrod operations in cohomology
• Provide additional structure on K-Theory groups
• Can be used to study the torsion in K-Theory
• Example: The Adams operations can detect the non-triviality of certain vector bundles over spheres

Atiyah-Hirzebruch spectral sequence for K-Theory

Definition and construction

• The Atiyah-Hirzebruch spectral sequence (AHSS) is a spectral sequence that converges to the K-Theory groups of a space X
• Relates the K-Theory of X to its ordinary cohomology
• The E_2 page of the AHSS is given by E_2^{p,q} = H^p(X; K^q(point)), where H^p denotes ordinary cohomology and K^q(point) is the K-Theory of a point in degree q

Differentials and torsion

• The differentials in the AHSS are related to the Steenrod operations in cohomology
• Provide a way to determine the torsion in the K-Theory groups
• Example: The differentials in the AHSS can detect the torsion in the K-Theory of lens spaces

Computation of K-Theory groups

• To calculate the K-Theory groups using the AHSS:
  1. Compute the E_2 page using the cohomology of X and the known values of K^q(point)
  2. Determine the differentials and compute the subsequent pages of the spectral sequence
  3. The K-Theory groups are given by the limit of the spectral sequence, i.e., K^n(X) = lim_{p+q=n} E_\infty^{p,q}
• The AHSS is particularly useful for computing the K-Theory of spaces with known cohomology (spheres, projective spaces, certain homogeneous spaces)
• Example: The AHSS can be used to compute the K-Theory of the complex Grassmannian, which is a quotient of the unitary group