nLab
splitting principle

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Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Chern-Weil theory

Contents

Idea

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

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

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

f *:H (BG)H (BT n), f^\ast \;\colon\; H^\bullet(B G) \hookrightarrow H^\bullet(B T^n) \,,

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

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

Let XX be a connected topological space and PXP \to X a GG-principal bundle over XX classified by a map g:XBGg \colon X \to B G.

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

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

This diagram shows that the pullback of the GG-principal bundle PXP \to X along pp to YY is equivalently a TT-principal bundle splitting as circle group-principal bundles classified by (g 1,,g n)(g_1, \cdots, g_n).

That this is a useful splitting is the content of:

Theorem

(generalized splitting principle)

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

Then

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

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

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

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

    p *:H (X,R)H (Y,R). 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

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

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

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

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

Remark

One way to see that Bi *:H *(BG)H *(BT)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 *𝔤 Sym *𝔱 Sym^{\ast} \mathfrak{g}^{\vee} \to Sym^{\ast} \mathfrak{t}^{\vee} for GG a compact Lie group. This tells us firstly that these cohomology rings are particularly nice.

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

Examples

Complex vectors bundles and their Chern roots

For G=U(n)G = U(n) the unitary group, the universal characteristic classes are the Chern classes c kH (BU(n),)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

(Bi) *( kc k)=(1+x 1)(1+x n), (B i)^\ast(\sum_k c_k) = (1 + x_1) \cdots (1+ x_n) \,,

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

c k(p *ξ)=σ k(c 1(ζ 1),,c n(ζ n)). c_k(p^\ast \xi ) = \sigma_k(c_1(\zeta_1), \cdots, c_n(\zeta_n)) \,.

This case is what is traditionally is often meant by default by 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 used 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 ϕK_\phi is a series in a single variable x=c 1(L)x = c_1(L), to be thought of as the first Chern class of the universal complex line bundle over BU(1)B U(1).

The Hirzebruch formula for the value of the genus ϕ\phi on an oriented manifold XX

ϕ(X)=K ϕ(TX),[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 kK ϕ(x k)\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

See also

Discussion in the context of complex oriented cohomology theory and their generalized Chern classes includes

More expository discussion in the context of characteristic classes with applications in mathematical physics is in

  • Yang Zhang, A brief introduction to characteristic classes

    from the differentiable viewpoint_, 2011 (pdfrief introduction to characteristic classes from the differentiable viewpoint.pdf))

The generalization to a splitting principle for twisted vector bundles (twisted cohomology) is discussed (in terms of bundle gerbe modules) in

  • Atsushi Tomoda, On the splitting principle of bundle gerbe modules, Osaka J. Math. Volume 44, Number 1 (2007), 231-246. (Euclid, talk slides pdf)

For characteristic classes of linear representations:

Last revised on January 28, 2019 at 03:11:25. See the history of this page for a list of all contributions to it.