5.1 Definition and properties of Chern classes

Chern classes are key tools in K-theory, measuring the twisting of complex vector bundles. They live in even-dimensional cohomology groups and have properties like naturality under pullbacks and stability under direct sums. These characteristics make them invaluable for studying bundle structures.

Understanding Chern classes is crucial for grasping the Chern character, which links K-theory to rational cohomology. This connection allows us to translate between K-theoretic and cohomological information, bridging two fundamental areas of algebraic topology.

Chern Classes for Vector Bundles

Definition and Basic Properties

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• Chern classes are characteristic classes associated to complex vector bundles that measure the twisting of the bundle
• For a complex vector bundle $E$ over a topological space $X$, the $i$-th Chern class $c_i(E)$ is an element of the $2i$-th cohomology group $H^{2i}(X; \mathbb{Z})$
• The total Chern class $c(E)$ is defined as the formal sum of all Chern classes: $c(E) = 1 + c_1(E) + c_2(E) + \cdots$
• Chern classes are natural under pullbacks: if $f: Y \to X$ is a continuous map and $E$ is a complex vector bundle over $X$, then $c_i(f^*(E)) = f^*(c_i(E))$, where $f^*$ denotes the induced map on cohomology
• Chern classes are stable under direct sums: if $E$ and $F$ are complex vector bundles over $X$, then $c(E \oplus F) = c(E) \smile c(F)$, where $\smile$ denotes the cup product in cohomology

Computation and Examples

• Trivial bundle: For the trivial bundle $X \times \mathbb{C}^n$ over a space $X$, all Chern classes vanish except for $c_0$, which equals $1$
• Line bundles: For a line bundle $L$, the only non-trivial Chern class is $c_1(L)$, which equals the Euler class of $L$ (e.g., the tautological line bundle over $\mathbb{CP}^1$)
• Tangent bundle of $\mathbb{CP}^n$: The total Chern class of the tangent bundle $T\mathbb{CP}^n$ is given by $c(T\mathbb{CP}^n) = (1 + x)^{n+1}$, where $x$ is the generator of $H^2(\mathbb{CP}^n; \mathbb{Z})$
• Tautological bundles over Grassmannians: The Chern classes of the tautological bundle over the Grassmannian $Gr(k, n)$ can be expressed in terms of Schur polynomials

Axioms and Properties of Chern Classes

Axioms

• Normalization: For the tautological line bundle $\gamma^1$ over the complex projective space $\mathbb{CP}^{\infty}$, $c_1(\gamma^1)$ is the generator of $H^2(\mathbb{CP}^{\infty}; \mathbb{Z}) \cong \mathbb{Z}$
• Functoriality: Chern classes are natural under pullbacks, as described in the previous section
• Whitney sum formula: Chern classes are stable under direct sums, as described in the previous section
• Nilpotency: For a complex vector bundle $E$ of rank $r$, the Chern classes $c_i(E)$ vanish for $i > r$

Splitting Principle and Flag Bundles

• Splitting principle: Every complex vector bundle $E$ over a paracompact Hausdorff space $X$ can be pulled back from the direct sum of line bundles over the flag bundle of $E$
• The flag bundle $F(E)$ is a space whose points are complete flags in the fibers of $E$, and there is a natural projection $\pi: F(E) \to X$
• Over $F(E)$, the pullback bundle $\pi^*(E)$ splits as a direct sum of line bundles, which allows for the computation of Chern classes using the Whitney sum formula

Chern Classes in Examples

Chern Classes of Projective Spaces

• Complex projective spaces $\mathbb{CP}^n$ are fundamental examples in the study of Chern classes
• The cohomology ring of $\mathbb{CP}^n$ is given by $H^*(\mathbb{CP}^n; \mathbb{Z}) \cong \mathbb{Z}[x]/(x^{n+1})$, where $x = c_1(\gamma^1)$ is the generator of $H^2(\mathbb{CP}^n; \mathbb{Z})$
• The total Chern class of the tangent bundle $T\mathbb{CP}^n$ is $c(T\mathbb{CP}^n) = (1 + x)^{n+1}$, which can be expanded to compute individual Chern classes

Chern Classes of Tautological Bundles

• Over the Grassmannian $Gr(k, n)$ of $k$-dimensional subspaces of $\mathbb{C}^n$, there is a tautological bundle $\gamma^k$ whose fiber over a point $[V] \in Gr(k, n)$ is the subspace $V$ itself
• The Chern classes of $\gamma^k$ can be expressed using Schur polynomials in the Chern classes of the universal bundle over the classifying space $BU(k)$
• These expressions are related to the Schubert calculus on Grassmannians and have applications in enumerative geometry

Geometric and Topological Interpretation of Chern Classes

Obstruction Theory and Triviality

• The vanishing of all Chern classes of a complex vector bundle is a necessary (but not sufficient) condition for the bundle to be trivial
• Chern classes can be interpreted as obstructions to the existence of certain geometric structures on the bundle (e.g., a global frame or a flat connection)
• The non-vanishing of Chern classes indicates the presence of topological twisting in the bundle, which prevents it from being trivial

Chern Classes and Curvature

• In the context of Hermitian vector bundles with connection, Chern classes can be represented by differential forms involving the curvature tensor
• The $i$-th Chern class $c_i(E)$ is represented by a closed differential form of degree $2i$ that is a polynomial in the curvature forms of the bundle
• This relationship between Chern classes and curvature is a key aspect of the Chern-Weil theory, which connects the topological and geometric properties of vector bundles

Chern Numbers and Characteristic Numbers

• For a compact complex manifold $M$, the integrals of products of Chern classes over $M$, known as Chern numbers, are important invariants that capture topological and geometric information about $M$
• Chern numbers are examples of characteristic numbers, which are topological invariants obtained by integrating characteristic classes (such as Chern classes) over the manifold
• The Gauss-Bonnet-Chern theorem expresses the Euler characteristic of a compact Riemannian manifold $M$ as the integral of a certain combination of Chern classes of the tangent bundle $TM$

Chern Character and K-Theory

• The Chern character is a ring homomorphism from the K-theory of a space to its rational cohomology, which can be expressed in terms of Chern classes
• For a complex vector bundle $E$, the Chern character $ch(E)$ is defined as $ch(E) = \sum_{i=0}^{\infty} \frac{1}{i!} c_1(E)^i$, where $c_1(E)^i$ denotes the $i$-th power of the first Chern class in cohomology
• The Chern character provides a connection between K-theory and ordinary cohomology, allowing for the computation of K-theoretic invariants using Chern classes and vice versa