9.2 Chern classes and Stiefel-Whitney classes

Chern classes and Stiefel-Whitney classes are key tools for understanding vector bundles. They measure how twisted or non-trivial a bundle is, providing crucial information about its structure and properties.

These characteristic classes have applications in geometry and topology. They help determine when bundles are trivial and provide obstructions to certain structures on manifolds, connecting abstract algebra to concrete geometric insights.

Chern Classes for Complex Bundles

Definition and Properties

• Chern classes are characteristic classes associated with complex vector bundles, providing a way to measure the twisting and non-triviality of the bundle
• The i-th Chern class $c_i(E)$ of a complex vector bundle $E$ is an element of the 2i-th cohomology group $H^{2i}(B; Z)$, where $B$ is the base space of the bundle
• The total Chern class $c(E)$ is defined as the formal sum of all Chern classes: $c(E) = 1 + c_1(E) + c_2(E) + ... + c_n(E)$, where $n$ is the rank of the vector bundle
• Chern classes are natural under pullbacks, meaning that if $f: X \to B$ is a continuous map and $E$ is a complex vector bundle over $B$, then $f^*(c_i(E)) = c_i(f^*(E))$, where $f^*(E)$ is the pullback bundle
• The first Chern class $c_1(E)$ is also known as the Euler class when $E$ is a complex line bundle (rank 1)

Computation and Applications

• Chern classes can be computed using the splitting principle, which states that every complex vector bundle can be pulled back to a sum of line bundles, and the Chern classes of the original bundle are the elementary symmetric polynomials in the first Chern classes of the line bundles
• The vanishing of all Chern classes is a necessary and sufficient condition for a complex vector bundle to be trivial
• Chern classes provide obstructions to the existence of certain geometric structures on manifolds, such as almost complex structures and complex structures
• The Chern character is a ring homomorphism from the K-theory of a space to its rational cohomology, defined in terms of Chern classes, which provides a connection between K-theory and ordinary cohomology
• Example: The complex projective space $\mathbb{CP}^n$ has a tautological line bundle $\gamma^1$ whose first Chern class generates the cohomology ring of $\mathbb{CP}^n$

Properties of Chern Classes

Invariance and Whitney Sum Formula

• Chern classes are invariant under bundle isomorphisms, meaning that isomorphic bundles have the same Chern classes
• The Whitney sum formula relates the Chern classes of a direct sum of vector bundles to the Chern classes of the individual bundles: $c(E \oplus F) = c(E) \smile c(F)$, where $\smile$ denotes the cup product
• Example: If $E$ and $F$ are complex line bundles, then $c_1(E \oplus F) = c_1(E) + c_1(F)$

Relation to Euler Class and Stiefel-Whitney Classes

• The top Chern class $c_n(E)$ of a rank $n$ complex vector bundle $E$ is equal to the Euler class of the underlying oriented real vector bundle of rank $2n$
• For a complex vector bundle $E$, the mod 2 reduction of the total Chern class $c(E)$ is equal to the total Stiefel-Whitney class of the underlying real vector bundle: $\rho_2(c(E)) = w(E_R)$, where $\rho_2$ is the mod 2 reduction homomorphism and $E_R$ is the underlying real vector bundle of $E$
• The odd Stiefel-Whitney classes of the underlying real vector bundle of a complex vector bundle are all zero: $w_{2i+1}(E_R) = 0$ for all $i$
• The even Stiefel-Whitney classes of the underlying real vector bundle of a complex vector bundle are the mod 2 reductions of the Chern classes: $w_{2i}(E_R) = \rho_2(c_i(E))$ for all $i$

Stiefel-Whitney Classes for Real Bundles

Definition and Properties

• Stiefel-Whitney classes are characteristic classes associated with real vector bundles, analogous to Chern classes for complex vector bundles
• The i-th Stiefel-Whitney class $w_i(E)$ of a real vector bundle $E$ is an element of the i-th cohomology group $H^i(B; \mathbb{Z}/2\mathbb{Z})$, where $B$ is the base space of the bundle and $\mathbb{Z}/2\mathbb{Z}$ is the field of two elements
• The total Stiefel-Whitney class $w(E)$ is defined as the formal sum of all Stiefel-Whitney classes: $w(E) = 1 + w_1(E) + w_2(E) + ... + w_n(E)$, where $n$ is the rank of the vector bundle
• Stiefel-Whitney classes are natural under pullbacks, similar to Chern classes
• The first Stiefel-Whitney class $w_1(E)$ classifies the orientability of the vector bundle $E$: the bundle is orientable if and only if $w_1(E) = 0$
• The top Stiefel-Whitney class $w_n(E)$ of a rank $n$ real vector bundle $E$ is equal to the mod 2 Euler class of the bundle

Computation and Applications

• The Whitney product formula for Stiefel-Whitney classes is analogous to the one for Chern classes: $w(E \oplus F) = w(E) \smile w(F)$, where $\smile$ denotes the cup product
• The vanishing of all Stiefel-Whitney classes is a necessary and sufficient condition for a real vector bundle to be trivial, similar to the case of Chern classes for complex vector bundles
• Stiefel-Whitney classes provide obstructions to the existence of certain geometric structures on manifolds, such as spin structures and pin structures
• Example: The real projective space $\mathbb{RP}^n$ has a tautological line bundle $\gamma^1_{\mathbb{R}}$ whose first Stiefel-Whitney class generates the cohomology ring of $\mathbb{RP}^n$

Chern vs Stiefel-Whitney Classes

Relationship between Chern and Stiefel-Whitney Classes

• For a complex vector bundle $E$, the mod 2 reduction of the total Chern class $c(E)$ is equal to the total Stiefel-Whitney class of the underlying real vector bundle: $\rho_2(c(E)) = w(E_R)$, where $\rho_2$ is the mod 2 reduction homomorphism and $E_R$ is the underlying real vector bundle of $E$
• The odd Stiefel-Whitney classes of the underlying real vector bundle of a complex vector bundle are all zero: $w_{2i+1}(E_R) = 0$ for all $i$
• The even Stiefel-Whitney classes of the underlying real vector bundle of a complex vector bundle are the mod 2 reductions of the Chern classes: $w_{2i}(E_R) = \rho_2(c_i(E))$ for all $i$

Similarities and Differences

• Both Chern and Stiefel-Whitney classes are characteristic classes associated with vector bundles, providing a way to measure the twisting and non-triviality of the bundle
• Chern classes are defined for complex vector bundles, while Stiefel-Whitney classes are defined for real vector bundles
• Chern classes take values in integer cohomology, while Stiefel-Whitney classes take values in mod 2 cohomology
• The vanishing of all Chern or Stiefel-Whitney classes is a necessary and sufficient condition for a vector bundle to be trivial
• Both Chern and Stiefel-Whitney classes satisfy naturality under pullbacks and have similar Whitney sum formulas
• Example: The tangent bundle of the complex projective space $\mathbb{CP}^n$ has non-zero Chern classes, while the tangent bundle of the real projective space $\mathbb{RP}^n$ has non-zero Stiefel-Whitney classes