Mathematics.

algebraic topology

Characteristic Classes

Topology130 minDifficulty9 out of 10

You should know: vector bundles, cohomology

Overview

Characteristic classes are cohomology classes naturally associated to vector bundles that measure their twisting or non-triviality. The three main families are: Stiefel-Whitney classes w_i in H^i(B; Z/2) for real bundles, Chern classes c_i in H^{2i}(B; Z) for complex bundles, and Pontryagin classes p_i in H^{4i}(B; Z) for real bundles viewed via their complexification. They are the primary algebraic invariants distinguishing vector bundles and are central to the Atiyah-Singer index theorem.

Intuition

A trivial bundle has trivial characteristic classes. If a characteristic class is nonzero, the bundle cannot be trivial. Think of characteristic classes as 'fingerprints' of a bundle: they live in the cohomology of the base and encode global twisting data. Chern classes, for instance, can be thought of as measuring how many times the fibers wind around as you traverse loops in the base.

Formal Definition

Definition

A characteristic class for rank-k bundles with structure group G is a natural transformation from the functor Vect_k(-) (isomorphism classes of rank-k bundles) to the cohomology functor H^*(-, R). Equivalently, it is a cohomology class in H^*(BG; R) pulled back along the classifying map. The total Chern class of a complex rank-k bundle E is c(E) = 1 + c_1(E) + ... + c_k(E) in H^*(B; Z), defined axiomatically.

c(E)=1+c1(E)+c2(E)+⋯+ck(E)∈H∗(B;Z)c(E) = 1 + c_1(E) + c_2(E) + \cdots + c_k(E) \in H^*(B;\mathbb{Z})
Total Chern class
c(E⊕F)=c(E)⋅c(F)c(E\oplus F) = c(E)\cdot c(F)
Whitney product formula
c1(L)=e(LR)∈H2(B;Z)c_1(L) = e(L_\mathbb{R}) \in H^2(B;\mathbb{Z})
First Chern class = Euler class of underlying real bundle
w(E)=1+w1(E)+⋯+wk(E)∈H∗(B;Z/2)w(E) = 1 + w_1(E) + \cdots + w_k(E) \in H^*(B;\mathbb{Z}/2)
Total Stiefel-Whitney class

Theorems

Theorem 1: Axioms for Chern Classes
Chernclassesci(E)∈H2i(B;Z)satisfy:(1)c0=1;(2)naturalityf∗ci(E)=ci(f∗E);(3)Whitneyproductc(E⊕F)=c(E)c(F);(4)c1(γ1)isthecanonicalgeneratorofH2(CP∞;Z)forthetautologicalbundle.Chern classes c_i(E) \in H^{2i}(B;\mathbb{Z}) satisfy: (1) c_0 = 1; (2) naturality f^*c_i(E) = c_i(f^*E); (3) Whitney product c(E\oplus F) = c(E)c(F); (4) c_1(\gamma^1) is the canonical generator of H^2(\mathbb{CP}^\infty;\mathbb{Z}) for the tautological bundle.
Theorem 2: Non-immersion Theorem (Whitney)
IftheStiefel−WhitneyclassesofthestablenormalbundleofMnarenontrivialabovedegreek,thenMcannotbeimmersedinRn+k.If the Stiefel-Whitney classes of the stable normal bundle of M^n are nontrivial above degree k, then M cannot be immersed in \mathbb{R}^{n+k}.
Theorem 3: Chern-Weil Theory
ForacomplexbundleEwithconnectionAandcurvatureFA,theChernclassescanberepresentedbythecloseddifferentialformsck(E)=[det⁡(I+i2πFA)k].For a complex bundle E with connection A and curvature F_A, the Chern classes can be represented by the closed differential forms c_k(E) = [\det(I + \frac{i}{2\pi}F_A)_k].

Worked Examples

  1. The tautological bundle gamma^1 over CP^1 has total space {([z], w) in CP^1 × C^2 : w is a scalar multiple of z}.

  2. Since gamma^1 is a complex line bundle (rank 1), only c_0 and c_1 can be nonzero.

    c(γ1)=1+c1(γ1)∈H∗(CP1;Z).c(\gamma^1) = 1 + c_1(\gamma^1) \in H^*(\mathbb{CP}^1;\mathbb{Z}).

  3. By axiom (4), c_1(gamma^1) is the negative of the canonical generator of H^2(CP^1; Z) = Z.

    c1(γ1)=−[ω]∈H2(CP1;Z)≅Z,c_1(\gamma^1) = -[\omega] \in H^2(\mathbb{CP}^1;\mathbb{Z}) \cong \mathbb{Z},

  4. where [omega] is the fundamental class (generator). So c(gamma^1) = 1 - h, where h is the hyperplane class.

Answer: c(gamma^1) = 1 - h in H^*(CP^1; Z), where h is the positive generator of H^2.

Practice Problems

Difficulty 7/10

Show that the total Chern class of the direct sum of two line bundles L_1 ⊕ L_2 is (1 + c_1(L_1))(1 + c_1(L_2)).

Difficulty 9/10

Prove that if E -> B is a trivial rank-k vector bundle, then all its Chern classes c_i(E) = 0 for i >= 1.

Difficulty 8/10

Compute w_1 and w_2 of the tangent bundle T(RP^n) using the fact that w(T(RP^n)) = (1+a)^{n+1} in H^*(RP^n; Z/2).

Common Mistakes

Common Mistake

If all Chern classes vanish, the bundle is trivial.

Vanishing Chern classes is necessary but not sufficient for triviality. A bundle can have all zero Chern classes yet still be nontrivial (e.g., over higher-dimensional base spaces with torsion in cohomology).

Common Mistake

Pontryagin classes and Chern classes are the same for real bundles.

Pontryagin classes p_i(E) = (-1)^i c_{2i}(E ⊗ C) for a real bundle E, so they are the Chern classes of the complexification — not the same as the Chern classes of E (which don't make sense for real bundles).

Quiz

In which cohomology group do the Chern classes c_i of a complex bundle live?
What does the Whitney product formula state?
Which characteristic class takes values in H^*(B; Z/2)?

Summary

  • Characteristic classes are cohomological invariants of vector bundles: Stiefel-Whitney (Z/2 coefficients), Chern (Z, for complex bundles), and Pontryagin (Z, for real bundles).
  • They are defined axiomatically by naturality, the Whitney product formula, and normalization on the tautological bundle.
  • The vanishing of a characteristic class is necessary (but not sufficient) for a bundle to be trivial.
  • Chern-Weil theory realizes characteristic classes as differential forms built from the curvature of a connection.
  • Applications include obstruction theory, the non-immersion theorem, and the Atiyah-Singer index theorem.

References

  1. BookMilnor, J. & Stasheff, J. — Characteristic Classes. Princeton University Press, 1974.
  2. WebsiteWikipedia — Characteristic class