This course develops the theory of characteristic classes from the curvature of connections on fibre bundles, showing how geometric data on a bundle produce topological invariants of the underlying manifold. The central goal is to explain why curvature is not just a local analytic quantity, but also a generator of cohomology classes that detect global bundle structure. Along the way, the course treats complex and real vector bundles, oriented bundles, and the special role of characteristic numbers as computable global measurements.

The chapters build from the basic passage from curvature to cohomology, then construct the Chern-Weil homomorphism as the main machine translating invariant polynomials into closed differential forms. From there, the course studies the standard characteristic classes: Chern classes for complex bundles, the Euler class via Pfaffian curvature, and Pontryagin classes for real bundles. Later chapters move from primary classes to secondary and transgressive phenomena, including Chern-Simons forms and invariants, and then connect these ideas to local index-theoretic forms and explicit geometric computations in representative bundles.

The final chapters emphasize applications and interpretation. They examine characteristic classes as obstructions to geometric structures and as numerical invariants extracted from manifolds, with special attention to four-dimensional geometry and instanton examples. The course closes by synthesizing the subject into a coherent picture: characteristic classes are the bridge between local differential geometry and global topology, encoding both hidden obstructions and measurable geometric content.

# Introduction

This course studies a central bridge in differential geometry: curvature is local and analytic, while characteristic classes are global and topological. The guiding question is how a differential form built from the curvature of a connection can define a cohomology class that does not depend on the connection. The answer is the Chern-Weil construction, which turns invariant polynomials on Lie algebras into characteristic classes of principal and vector bundles.

The preceding fibre bundles material supplied principal bundles, associated vector bundles, and connections. This chapter fixes the viewpoint for the course: connections are not only devices for parallel transport, but also a calculus for producing cohomological invariants. Chapters 1 and 2 make the Chern-Weil mechanism precise, Chapters 3 through 6 specialise it to Chern, Euler, Pontryagin, and characteristic-number calculations, and Chapters 7 and 8 study transgression and Chern-Simons forms as secondary invariants measuring choices of connection.

## The Organising Problem

A vector bundle or principal bundle can be described locally by transition functions, but characteristic classes should detect global twisting in a way that survives changes of trivialisation. Curvature forms are defined after choosing a connection, so the first issue is to understand why they can encode topological information rather than only auxiliary geometric data.

[explanation: Local Curvature And Global Invariants] Let $P \to M$ be a principal $G$-bundle over a smooth manifold $M$, and let $A$ be a connection on $P$ with curvature $F_A$. The curvature is a horizontal, equivariant $\frak g$-valued $2$-form on $P$, or equivalently an adjoint-bundle-valued $2$-form on $M$. If $q$ is a polynomial on $\frak g$ invariant under the adjoint action of $G$, then $q(F_A)$ descends to an ordinary differential form on $M$.

The two structural facts behind the course are that $q(F_A)$ is closed and that its de Rham cohomology class is independent of $A$. Closedness is a differential consequence of the Bianchi identity. Independence from the connection is obtained by comparing two connections along a path and writing the derivative of the resulting forms as an exact form. [/explanation]

The construction needs invariant polynomials because curvature changes by conjugation under changes of local frame. A polynomial expression that is not invariant would depend on the chosen trivialisation and therefore would not define a differential form on the base. For instance, on a rank two vector bundle the function sending a matrix to its $(1,1)$-entry changes after conjugating the curvature matrix by a change-of-frame matrix, while $\operatorname{tr}(X)$ and $\operatorname{tr}(X^2)$ are unchanged by conjugation.

[definition: Invariant Polynomial] Let $G$ be a Lie group with [Lie algebra](/page/Lie%20Algebra) $\frak g$. An invariant polynomial of degree $k$ on $\frak g$ is a [homogeneous polynomial](/page/Homogeneous%20Polynomial) map $q:\frak g \to \bb R$ or $q:\frak g \to \bb C$ such that \begin{align*} q(\operatorname{Ad}_g X) = q(X) \end{align*} for all $g \in G$ and $X \in \frak g$. [/definition]

After polarisation, such a polynomial is often treated as a symmetric $k$-linear form on $\frak g$ that is invariant under the adjoint action in every argument. This is the form in which it is evaluated on curvature forms.

[example: Trace Polynomials] Let $E \to M$ be a complex vector bundle with a connection whose curvature is represented in one local frame by a matrix-valued $2$-form $F$. If the local frame is changed by an invertible matrix-valued function $g$, then the curvature matrix in the new frame is $F' = g^{-1}Fg$. For an ordinary matrix $X$ and $k \ge 1$, the trace polynomial $q_k(X)=\operatorname{tr}(X^k)$ is invariant under conjugation because \begin{align*} (g^{-1}Xg)^2 = g^{-1}Xgg^{-1}Xg = g^{-1}X^2g. \end{align*} Repeating the same cancellation $gg^{-1}=I$ gives \begin{align*} (g^{-1}Xg)^k = g^{-1}X^kg. \end{align*} Therefore \begin{align*} \operatorname{tr}((g^{-1}Xg)^k)=\operatorname{tr}(g^{-1}X^kg). \end{align*} Using cyclicity of trace, $\operatorname{tr}(AB)=\operatorname{tr}(BA)$, with $A=g^{-1}$ and $B=X^kg$, we get \begin{align*} \operatorname{tr}(g^{-1}X^kg)=\operatorname{tr}(X^kgg^{-1})=\operatorname{tr}(X^k). \end{align*} The same algebra applies to the matrix-valued curvature form, with matrix multiplication combined with wedge product. Since $F$ is a $2$-form, it commutes in sign with the degree-$0$ matrix entries of $g$ and $g^{-1}$, so \begin{align*} (F')^k=(g^{-1}Fg)^k=g^{-1}F^kg. \end{align*} Taking traces gives \begin{align*} \operatorname{tr}((F')^k)=\operatorname{tr}(g^{-1}F^kg)=\operatorname{tr}(F^kgg^{-1})=\operatorname{tr}(F^k). \end{align*} Thus the local forms $\operatorname{tr}(F^k)$ agree on overlaps and patch to a globally defined $2k$-form on $M$. These trace forms are the basic curvature expressions from which Chern character forms are built. [/example]

This example explains why matrix operations appear throughout the subject. The topological invariant is not the curvature matrix itself, since that matrix changes with the frame, but a conjugation-invariant expression in that matrix.

## Connections, Curvature, And Characteristic Forms

The immediate problem is to turn a connection-dependent curvature form into a closed ordinary form on the base. The course uses the same mechanism repeatedly: choose a connection, apply an invariant polynomial to its curvature, and then prove that the resulting de Rham class is independent of the choice.

[definition: Characteristic Form] Let $P \to M$ be a principal $G$-bundle with connection $A$ and curvature $F_A$. Let $q$ be an invariant polynomial of degree $k$ on $\frak g$. The characteristic form associated to $(q,A)$ is the differential form $q(F_A) \in \Omega^{2k}(M)$ obtained by evaluating the polarised form associated to $q$ on $F_A,\dots,F_A$. [/definition]

The notation suppresses the descent from $P$ to $M$. In local trivialisations it is computed by applying $q$ to the local curvature matrix or Lie-algebra-valued curvature form. For this construction to enter de Rham cohomology, the form must be closed; the next theorem proves exactly this property from the Bianchi identity and the adjoint-invariance built into $q$.

[quotetheorem:9757]