Zoran Skoda hom11lec23

Mon, March 26, 17 o’clock. Coursepage. Next lecture 24. Previous lectures 1,2,3,4,5,6,7,8,9,10,11,12, 13,14,15,16,17,18,19,20,21,22,zadaci.

This lecture is dedicated to Chern-Weil theory of characteristic classes of vector bundles or principal bundles in de Rham cohomology theory (cf. de Rham complex?).

Characteristic classes in general

If $H$ is some cohomology theory, a characteristic class of vector bundles (or of principal bundles) is the assignment which to every bundle $\xi = (E\stackrel{\pi}\longrightarrow M)$ over a paracompact manifold $M$ gives some cohomology class $c(\xi)\in H^*(M)$ of the base manifold, in a way which commutes with pullback: for every continuous $f: N\to M$ and $\xi$ over $M$, the cohomology classes $c(f^*\xi)=f^*(c(\xi))$ are equal, where on the left we have a pullback of bundles and on the right-hand side the pullback in cohomology ($f^* = H(f)$). In other words, a characteristic class is a functorial assignment of a cohomology class of the base to the bundle over a base. In homotopy theory it is known that for every topological group $G$, the functor which assigns to each paracompact Hausdorff space $X$ the set of isomorphism classes of vector bundles over $X$ and which to a continuous map $f:X\to Y$ assigns the pullback functor (at the level of isomorphism classes) is representable in homotopy category; the representing object (is whose homotopy type is) called the classifying space $B G$. in special cases there are models of the homotopy type of $B G$. For example, for the closed subgroups of $GL(n)$ there are models which are inductive limits of smooth manifolds of increasing dimension, e.g. of Grassmanians, and for the discrete groups there is the Milnor model using the infinite join. The general result is the application of Brown’s representability theorem which says that a contravariant functor from the homotopy category of connected CW-complexes to sets is representable provided it has some exactness properties (the wedge and Mayer-Vietoris axioms). Brown’s theorem is usually proven in general by induction on the dimension of the cells, inserting more cells to improve the $(n-1)$-representability to $n$-representability (typically using the machinery of simplicial approximation lemma); consequently some choices at each induction step are possible and the result is a CW-complex which is determined only up to homotopy type.

The representability means that the set of homotopy classes of maps $[X,B G]$ is in a natural bijection with the set of isomorphism classes of bundles over $X$, namely there is a universal bundle $u$ over $B G$ such that for every bundle $\xi$ over $X$ there is a unique homotopy class $[f]$ of continuous maps $f:X\to B G$ such that $\xi = f^*(u)$. Now if $H$ is some cohomology theory and $c_0\in H^*(B G)$ some cohomology class, then the formula $c(\xi) := f^*_\xi (c_0)$, where $f_\xi$ is a representative of the homotopy class of maps $X\to B G$ corresponding to $\xi$, defines a characteristic class as a functor. It is simple to check that, if the classifying space $B G$ exists, then all characteristic classes in cohomology $H$ are of this form. Thus, sometimes we simply consider the elements of $H^*(B G)$ as the characteristic classes in cohomology $H$.

Chern-Weil theory

Chern-Weil theory is a recipe how to construct characteristic classes in de Rham cohomology of smooth bundles with help of connections. In Chern-Weil theory, every characteristic class $c = c^P$ is determined by an invariant polynomial $P$ on the Lie algebra $\mathfrak{g}$ of the structure group of the bundle. An invariant polynomial is a polynomial $P$ in elements of a $n\times n$-matrix with coefficients in the ground field with the property $P(A B) = P(B A)$ for any two matrices $A,B$ over the ground field. This is equivalent to have the property $P(C^{-1} A C) = P(A)$ for any $A$ and any regular matrix $C$. An invariant polynomial can be applied to a matrix depending on a parameter. The Chern-Weil recipe is as follows: for every bundle $\xi$ over $M$ construct some nonzero connection on $\xi$. Such a connection exists by the partition of unity arguments. Then compute its curvature. Curvature tensor is a globally defined 2-tensor $R$, but locally it is equivalent to the data of the curvature 2-form $\Omega$ (which is defined only in a local chart). Computing $P(\Omega)$ on a neighborhood where the curvature form $\Omega$ is defined in local coordinates gives a differential form of the total space of the bundle restricted on the neighborhood. This differential form does not depend on the local coordinates, as the curvature 2-form under change of coordinates gets conjugated by a fiber depending matrix hence applying the invariant polynomial on it does not change the result. One proves that the resulting differential form is closed, hence defines a de Rham cohomology class. One finally proves, using the homotopy argument, that this class does not depend on the chosen connection we started with, but only on the bundle.

Even degree only

Notice that the Chern-Weil theory gives the cohomology classes in even degree only. This is evident because the curvature is a 2-form so the invariant polynomial can in principle give only classes in the degree $2 k$, where $k$ is an integer or $0$. If one calculates the cohomology of the classifying space for $O(n)$ (which corresponds to the case of smooth $n$-dimensional bundles over Riemannian manifolds) then one sees that no characteristic classes (in strict sense, in de Rham cohomology) other than those obtained from Chern-Weil recipe exist. However, there are certain obstruction classes in odd degree which share a similar role, which are called the secondary characteristic clases, for example the class corresponding to the Chern-Simons functional in topological quantum field theory in 3 dimensions.

Examples of characteristic classes

Chern classes of complex bundles (over $\mathbb{C}$, but the class is in even degree, hence real), Pontrjagin classes of real vector bundles (in degree divisible by $4$), Stiefel-Whitney class (coefficients in $\mathbb{Z}_2$).

Total Chern class (sum of all Chern classes with the weight given by an unknown, in other words the generating function for the Chern classes), Todd class.