Contents

Contents

Idea

In algebraic topology, a splitting principle for a classifying space $B G$ for some topological group $G$ (delooping of some ∞-group $G$) is a map from a torus-classifying space

$B (T^n) \overset{f}{\longrightarrow} B G$

such that the induced pullback of cohomology rings is an injective function:

$f^\ast \;\colon\; H^\bullet(B G) \hookrightarrow H^\bullet(B T^n) \,,$

hence allowing to view the cohomology of $G$-principal ∞-bundles in terms of that of plain torus-principal bundles.

In particular for $G$ a classical compact Lie group such as the unitary group, the splitting principle holds and allows to express Chern classes of complex vector bundles as algebraic expressions in just first Chern classes of complex line bundles.

Statement

Let $G$ be a connected compact Lie group and write $U(1)^n \simeq T \stackrel{i}{\hookrightarrow} G$ for a maximal torus.

Let $X$ be a connected topological space and $P \to X$ a $G$-principal bundle over $X$ classified by a map $g \colon X \to B G$.

Then consider the coset space $G/T$ and the $G/T$-fiber bundle $Y\to X$ associated to $P$, this is equivalently the homotopy pullback in the diagram

$\array{ Y &\stackrel{(g_1,\cdots, g_n)}{\longrightarrow}& B T & \simeq B U(1)^n \\ \big\downarrow {}^{\mathrlap{p}} && \big\downarrow {}^{\mathrlap{B i}} \\ X &\stackrel{g}{\longrightarrow}& B G } \,.$

This diagram shows that the pullback of the $G$-principal bundle $P \to X$ along $p$ to $Y$ is equivalently a $T$-principal bundle splitting as circle group-principal bundles classified by $(g_1, \cdots, g_n)$.

That this is a useful splitting is the content of:

Theorem

(generalized splitting principle)

Let $R$ be a commutative ring in which a prime number $p$ is a unit if $H_\bullet(B G,\mathbb{Z})$ has a $p$-torsion subgroup.

Then

1. The $H^\bullet(B G,R)$-module $H^\bullet(B T, R)$ (via $B i^\ast$) is free on the cohomology of $G/T$:

$H^\bullet(B T,R)\simeq H^\bullet(B G,R) \otimes H^\bullet(G/T, R) \,;$
2. Analogously there is an isomorphism

$H^\bullet(Y,R) \simeq H^\bullet(X,R)\otimes H^\bullet(G/T,R)$

and hence $p^\ast$ is the canonical inclusion (and hence in particular is an injection)

$p^\ast \;\colon\; H^\bullet(X,R)\hookrightarrow H^\bullet(Y,R) \,.$

In this general form this is due to (May).

Remark

Since the elements

$c \in H^\bullet(B G,R)$

are the universal characteristic classes of $G$-principal bundles with coefficients in $R$ (hence by Chern-Weil theory the invariant polynomials of the Lie algebra $\mathfrak{g}$ if $R$ has characteristic-0), theorem gives the following way to express the characteristic classes of $G$-principal bundles on $X$ by tuples

$(c_1^1, \cdots, c_1^n) \coloneqq (B i)^\ast c$

of characteristic classes – hence first Chern classes – of just circle bundles (line bundles):

$p^\ast (c(P)) \simeq (g_1^\ast c_1^1, \cdots, g_n^\ast c_1^n) \,.$

(Since $p^\ast$ is injective, this is a genuine characterization of $c(P)$).

Remark

One way to see that $B i^*\colon H^*(B G) \to H^*(B T)$ is injective is by using Chern-Weil theory to recognise that this map is just $Sym^{\ast} \mathfrak{g}^{\vee} \to Sym^{\ast} \mathfrak{t}^{\vee}$ for $G$ a compact Lie group. This tells us firstly that these cohomology rings are particularly nice.

One can define a transfer map $\tau\colon H^*(B T) \to H^*(B G)$ as in this MO answer, and then show, following (Dupont 1978, chapter 8), that $\tau\circ B i^*\colon H^*(B G)\to H^*(B G)$ is multiplication by the Euler class $\chi(G/T)$. Thus if $\chi(G/T) \gt 0$ then $\tau\circ B i^*$ hence $B i^*$ is injective. One can calculate $\chi(G/T) = | N(T)/T |$, where $N(T)/T =: W_T$ is the Weyl group of the maximal torus $T$, using a Lefshetz fixed point-argument, giving the result.

Examples

Complex vectors bundles and their Chern roots

For $G = U(n)$ the unitary group, the universal characteristic classes are the Chern classes $c_k \in H^\bullet(B U(n), \mathbb{Z})$. By the discussion at Chern class – Properties – Splitting principle and Chern roots. the universal splitting principle here says that

$(B i)^\ast(\sum_k c_k) = (1 + x_1) \cdots (1+ x_n) \,,$

where the $x_i \in H^\bullet(B U(1)^n , \mathbb{Z})$ are the universal characteristic classes of the maximal torus, hence are $n$ incarnations of the universal first Chern class (equivalently: the weights of the group characters of $U(n)$). It follows that every complex vector bundle $\xi$ of rank $n$ over a space $X$ when pulled back to its flag space bundle decomposes as a direct sum of complex line bundles $\zeta_i$ and has Chern classes $c_k$ expressed in terms of the first Chern classes of these line bundles as

$c_k(p^\ast \xi ) = \sigma_k(c_1(\zeta_1), \cdots, c_n(\zeta_n)) \,.$

This case is typically the default meaning of the “splitting principle”.

For the special unitary group the situation is the same, only that here the splitting is into a sum of line bundles whose tensor product is constrained to be trivializable.

Linear representations and Brauer induction

The Brauer induction theorem may be regarded as the splitting principle for linear representations (Symonds 91), see also at characteristic classes of linear representations,

Real vector bundles

Under the Relation between Pontryagin classes and Chern classes the above translates into a splitting principle also for real vector bundles.

Genera and Hirzebruch characteristic series

The basic theorem of Hirzebruch series expresses genera via the splitting principle. The Hirzebruch characteristic series $K_\phi$ is a series in a single variable $x = c_1(L)$, to be thought of as the first Chern class of the universal complex line bundle over $B U(1)$.

The Hirzebruch formula for the value of the genus $\phi$ on an oriented manifold $X$

$\phi(X) = \langle K_\phi(T X), [X]\rangle$

denotes the pairing of that class of the tangent bundle with the fundamental class which under the splitting principle pulls back on the flag space bundle to the class $\prod_k K_\phi(x_k)$ of the corresponding direct sum of line bundles.

References

Discussion in the derivation of Chern classes and Stiefel-Whitney classes includes