Why are characteristic classes well-defined?

Question

In the definition of characteristic classes for a complex vector bundle $E$ ober a topological space $X$, we consider some space $X_S$ and a continuous map $p: X \rightarrow X'$ such that $E$ is the pullback of some vector bundle $F$ over $X'$ and $F$ splits as the sum of some line bundles $F_1,..., F_n$ over $X'$. Then for some formal power series $f$ one considers $f(c_1(F_1))\cdot...\cdot f(c_1(F_n))$ and pulls this back to $X$ to get a characteristic class. My question is: Why is this construction well-defined, i.e. why does it not depend on the choice of $X'$ and $p$?

Answer 1

One way to express the "splitting principle" is the following: let $BU(n)$ be the classifying space of $U(n)$, and $BT(n)$ be the classifying space of the torus $T(n) = S^1 \times \dots \times S^1$. There is a natural map $BT(n) \to BU(n)$ coming from the embedding $T(n) \to U(n)$. On the level of functors represented, this corresponds to the construction which sends $n$ line bundles ($BT(n)$ classifies an $n$-tuple of line bundles) to their direct sum ($BU(n)$ classifies $n$-dimensional vector bundles). Now, a characteristic class for complex vector bundles is the same thing as an element of $H^*(BU(n))$, while a "characteristic class" of line bundles is an element in $H^*(BT(n))$. One way of expressing the splitting principle is that the induced map $$H^*(BU(n)) \to H^*(BT(n))$$ is injective with image the symmetric polynomials in the canonical generators $x_1, \dots, x_n \in H^*(BT(n))$. (The elementary symmetric polynomials correspond to the Chern classes.) So, if you want to construct a characteristic class on complex vector bundles (which will be a polynomial in the Chern classes), then you might as well pretend that your vector bundle splits as a sum of line bundles (i.e., admits a reduction to a $T(n)$-bundle), and then just make sure that the construction is symmetric in the line bundles involved.

There is a general construction that works for compact connected Lie groups and maximal tori, at least for cohomology with $\mathbb{Q}$-coefficients. Namely, if $G$ is a compact Lie group, then, using Chern-Weil theory, one has a map from $(\mathrm{Sym} \mathfrak{g}^*)^G$ to $H^*(BG; \mathbb{R})$ (that is, to an invariant polynomial on the Lie algebra we can associate a characteristic class for $G$-bundles), and this will be an isomorphism. If $T \subset G$ is a maximal torus with Weyl group $W$ and Lie algebra $\mathfrak{h}$, then we have a restriction isomorphism $$(\mathrm{Sym} \mathfrak{g}^*)^G \to (\mathrm{Sym} \mathfrak{h}^* )^W $$ (by a theorem of Chevalley), which implies that the induced map $$ H^*(BG; \mathbb{R}) \to H^*(BT, \mathbb{R})^W$$ is an isomorphism. The analog of the splitting principle is thus that, if you want to define a characteristic class for principal $G$-bundles (in real cohomology), then you just have to define one for the maximal torus, provided that it's invariant under the Weyl group.

Another relevant result here is the Atiyah-Segal completion theorem, which identifies the $K$-theory of $BG$ as the completed representation ring of $G$ (this because, modulo torsion, K-theory is essentially the same as even-dimensional cohomology).