group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
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
such that the induced pullback of cohomology rings is an injective function:
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.
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
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:
(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
The $H^\bullet(B G,R)$-module $H^\bullet(B T, R)$ (via $B i^\ast$) is free on the cohomology of $G/T$:
Analogously there is an isomorphism
and hence $p^\ast$ is the canonical inclusion (and hence in particular is an injection)
In this general form this is due to (May).
Since the elements
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
of characteristic classes – hence first Chern classes – of just circle bundles (line bundles):
(Since $p^\ast$ is injective, this is a genuine characterization of $c(P)$).
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.
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
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
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.
The Brauer induction theorem may be regarded as the splitting principle for linear representations (Symonds 91), see also at characteristic classes of linear representations,
Under the Relation between Pontryagin classes and Chern classes the above translates into a splitting principle also for real vector bundles.
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 $X$
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.
Peter May, A note on the splitting principle, Topology and its Applications Volume 153, Issue 4, 1 November 2005, Pages 605-609 (pdf, doi:10.1016/j.topol.2005.02.007)
Marcelo Aguilar, Samuel Gitler, Carlos Prieto, section 10.2 of Algebraic topology from a homotopical viewpoint, Springer (2002) (toc pdf)
Michael Hopkins (notes by Akhil Mathew), Lecture 10 in: Spectra and stable homotopy theory, 2012 (pdf, pdf)
Discussion in the derivation of Chern classes and Stiefel-Whitney classes includes
See also
Wikipedia, Splitting principle
Notes on the splitting principle (pdf)
Johan L. Dupont, Curvature and Characteristic Classes, Lecture Notes in Mathematics, 640 (1978) doi:10.1007/BFb0065364.
Discussion in the context of complex oriented cohomology theory and their generalized Chern classes includes
Dai Tamaki, Akira Kono, Section 3.3 and Theorem 3.22 in: Generalized Cohomology, Translations of Mathematical Monographs, American Mathematical Society, 2006 (ISBN: 978-0-8218-3514-2)
Jacob Lurie, lecture 4 of Chromatic Homotopy Theory, 2010 (pdf)
More expository discussion in the context of characteristic classes with applications in mathematical physics is in
The generalization to a splitting principle for twisted vector bundles (twisted cohomology) is discussed (in terms of bundle gerbe modules) in
For characteristic classes of linear representations:
Last revised on January 25, 2021 at 18:05:50. See the history of this page for a list of all contributions to it.