A NOTE ON THE SPLITTING PRINCIPLE
J.P. MAY
Abstract.We offer a new perspective on the splitting principle. We give
an easy proof that applies to all classical types of vector bundles and *
*in fact
to Gbundles for any compact connected Lie group G. The perspective gives
precise calculational information and directly ties the splitting princi*
*ple to the
specification of characteristic classes in terms of classifying spaces.
In the algebraic topology proseminar at Chicago, a student, Nils Barth, asked
for the precise relationship between the splitting principle and the specificat*
*ion of
Chern classes in terms of maximal tori. This note gives the quick and more gene*
*ral
answer that popped to mind. It should be utterly standard, but it was new to me
and to other faculty in the audience. Certainly I have not seen it in print.
Let T = T nbe a maximal torus in a compact connected Lie group G of rank
n and let R be a commutative ring in which p is invertible for all primes p such
that H*(G; Z) has ptorsion. Classical results of Borel [3] determine these pri*
*mes
explicitly when G is simply connected and describe how to determine them in ter*
*ms
of the elementary abelian psubgroups of G in general. Other classical results*
* of
Borel [1] imply that H*(BG; R) is a polynomial ring over R on n even degree
generators. The information relevant here is just that H*(BG; R) is concentrated
in even degrees.
By the BottSamelson theorem [4], H*(G=T ; Z) has no torsion and is also con
centrated in even degrees, hence H*(G=T ; R) is a free Rmodule. Let EG be a
universal principal Gbundle and take BT = EG=T and BG = EG=G. Inclusion of
orbits gives a Gbundle p: BT ! BG with fiber G=T . Taking cohomology with
coefficients in R henceforward, we see immediately that the Serre spectral sequ*
*ence
of this bundle collapses to give
(1) H*(BT ) ~=H*(BG) H*(G=T )
as an H*(BG)module via p*. In particular, H*(BT ) is a free H*(BG)module.
Now let , be a Gbundle over a space X. For convenience, we assume that X
is path connected. Let , have classifying map f :X ! BG. Of course, we can
think of , as a principal Gbundle or as a Gbundle with fiber F for any Gspace
F . Construct the pullback diagram
(2) Y ___g_//_BT
q p
fflffl fflffl
X _f__//_BG.
____________
1991 Mathematics Subject Classification. 55R40, 55N99.
Key words and phrases. splitting principle, characteristic classes, classify*
*ing spaces.
1
2 J.P. MAY
Then q :Y ! X is a Gbundle with fiber G=T . The action of ss1(X) on H*(G=T )
is trivial, since it is the pullback of the action of ss1(BG) = 0, and the elem*
*ents of
H*(G=T ) are permanent cycles in the Serre spectral sequence of q because they *
*are
permanent cycles in the Serre spectral sequence of p. Therefore the Serre spect*
*ral
sequence of q collapses to give an isomorphism
(3) H*(Y ) ~=H*(X) H*(G=T ).
The edge homomorphism shows that
q*: H*(X) ! H*(Y ) ~=H*(X) H*(G=T )
is the canonical inclusion, x ! x 1. Some readers might prefer to use the
EilenbergMoore spectral sequence to obtain these conclusions.
The map p is the universal example for the reduction of the structural group *
*of
a Gbundle from G to T . The Gbundle q*, over Y is classified by f O q = p O g*
* and
therefore has a canonical reduction. We view this reduction of the structural g*
*roup
of q*, as a generalized splitting principle.
Theorem 1 (Generalized splitting principle). For a Gbundle , over X, there is
a Gbundle q :Y ! X with fiber G=T and a reduction of the structural group of
q*, to T such that H*(Y ) ~=H*(X) H*(G=T ) and q* is the canonical inclusion.
Reinterpreting the diagram (2), the map g classifies a T bundle i that is the
fiberwise product of ncircle bundles ii with classifying maps the coordinates *
*gi of
g :Y ! BT ~=(BT 1)n. The equality f O q = p O g says that q*, is the Gbundle
obtained by extending the structure group of i from T to G. If we know p* on
characteristic classes, then we can read off the characteristic classes of q*, *
*from
those of the circle bundles ii. That is, for an element ff in H*(BG), thought o*
*f as
a characteristic class,
(4) ff(q*,) = q*ff(,) = q*f*(ff) = (g1, . .,.gn)*p*(ff).
Provided that ss1(G) is a free Abelian group, similar arguments work in Kthe*
*ory,
using its Serre or EilenbergMoore spectral sequence (see e.g. [5, 6]). In part*
*icular,
with this restriction on ss1(G), q* is a monomorphism in Ktheory.
We describe the classical examples, referring the reader to [2] for backgroun*
*d.
The first three apply to any commutative ring R and also work with evident modi
fications in Ktheory. Write H*(BT ) as the polynomial algebra in the n canonic*
*al
generators xi. For example, viewing BS1 = CP1 as K(Z, 2), the xi are the funda
mental classes. Let oeidenote the ith elementary symmetric function in n variab*
*les.
Example 2. Take G to be U(n), T = T nto be the subgroup of diagonal matrices,
and the fiber F to be Cn. The universal Chern classes are characterized as the
unique elements ci of H*(BU(n)) such that
p*(ci) = oei(x1, . .,.xn).
We have a splitting of q*, into the sum of n complex line bundles ii, and
ci(q*,) = oei(c1(i1), . .,.c1(in)).
Example 3. Take G to be SU(n) U(n), T = T n1 T nto be the subgroup
of diagonal matrices of determinant 1, and F to be Cn. With the left square an
A NOTE ON THE SPLITTING PRINCIPLE 3
instance of (2), we have the evident commutative diagram
(5) Y __g__//_BT n1_i__//_BT n
q p p
fflffl fflffl fflffl
X __f_//_BSU(n)__j_//_BU(n).
Here j*(c1) = 0 and j*(ci) = ci in H*(BSU(n)) for i > 1. Via i*, we can identify
H*(BT n1) as P [x1, . .,.xn]=(oe1(x1, . .,.xn)). The interpretation is that a*
*n n
plane U(n)bundle , with a reduction of its structural group to SU(n) splits al*
*ong
q as the sum of n line bundles whose tensor product is the trivial line bundle.
Example 4. Take G to be Sp(n), T to be the subgroup of diagonal matrices with
complex entries, and F to be Hn. Observe that T U(n) Sp(n), where the
second inclusion is given by extension of scalars from C to H. One way to define
the symplectic characteristic classes ki is by
X
j*(ki) = (1)a+icacb,
a+b=2i
where j :BU(n) ! BSp(n) is the induced map of classifying spaces, and this is
equivalent to p*(ki) = oe2iin H*(BT ). Here, in the diagram (2), p factors thro*
*ugh j.
The interpretation of the diagram is that if , is a quaternionic nplane bundle*
*, then
q*, splits as the sum of n quaternionic line bundles that are obtained by exten*
*sion
of scalars from n complex line bundles ii. Moreover,
ki(q*,) = oe2i(c1(i1), . .,.c1(in)).
In the following example, we require 2 to be invertible in R.
Example 5. Take G to be SO(2n + ") where n 1 and " = 0 or " = 1. Take
T = T n~=SO(2)n embedded in G as n (2 x 2)matrices along the diagonal, with
a last diagonal entry 1 if " = 1. Then H*(BSO(2n)) is the polynomial algebra on
the Pontryagin classes Pi, 1 i < n and the Euler class O, where O2 = Pn, and
H*(BSO(2n + 1)) is the polynomial algebra on the Pi, 1 i n. Observe that
T U(n) SO(n), where the second inclusion is given by identifying Cn with
R2n, as usual. One way to define Pi and O is by
X
j*(Pi) = (1)a+icacb and j*(O) = cn,
a+b=2i
where j :BU(n) ! BSO(2n) is the induced map of classifying spaces, and this is
equivalent to p*(Pi) = oe2iand p*(O) = oen in H*(BT ). Here, in the diagram (2)*
*, p
factors through j. The interpretation of the diagram is that if , is an oriente*
*d real
(2n + ")plane bundle, then q*, splits as the sum of the realifications of n co*
*mplex
line bundles ii and, if " = 1, a trivial real line bundle. Moreover,
Pi(q*,) = oe2i(c1(i1), . .,.c1(in))
and, if " = 0,
O(q*,) = oe(c1(i1), . .,.c1(in)).
We could easily go on to consider the exceptional Lie groups or to consider
generalizations to Hspaces and in particular to pcompact Lie groups, with the*
*ir
maximal tori. We leave such examples and elaborations to the interested reader.
4 J.P. MAY
Using maximal 2tori and mod 2 cohomology, we can often obtain an analogous
splitting principle in terms of real line bundles. We illustrate in the case o*
*f the
orthogonal groups, where we have the following analogues of Examples 2 and 3.
We now take cohomology with coefficients in the field F2.
Example 6. Let O(1)n O(n) be the subgroup of diagonal matrices. Write
H*(BO(1)) as the polynomial algebra in the n canonical basis elements xi. Let
p: BO(1)n ! BO(n) be the evident bundle with fiber O(n)=O(1)n. The Stiefel
Whitney classes are the unique elements wi of H*(BO(n)) such that
p*(wi) = oei(x1, . .,.xn).
The Serre spectral sequence of p has trivial local coefficients and collapses a*
*t E2.
Let , be a real nplane bundle over X with classifying map f :X ! BO(n) and
form the pullback
g n
Y _____//BO(1)
q  p
fflffl fflffl
X __f__//BO(n).
The Serre spectral sequence of q collapses at E2 to give an isomorphism
H*(Y ) ~=H*(X) H*(O(n)=O(1)n),
and q*: H*(X) ! H*(Y ) is the canonical inclusion, x ! x 1. We have a
splitting of q*, into the sum of n real line bundles ii, and
wi(q*,) = oei(w1(i1), . .,.w1(in)).
Example 7. Let O(1)n1 O(1)n be the subgroup of SO(n) O(n) consisting
of the diagonal matrices of determinant 1. Let , be an oriented real nplane bu*
*ndle
with classifying map f :X ! BSO(n) and define Y to be the pullback in the left
square of the diagram
g n1 i n
Y ____//_BO(1) _____//BO(1)

q p p
fflffl fflffl fflffl
X __f__//_BSO(n)__j__//BO(n).
Again, we have H*(Y ) ~= H*(X) H*(SO(n)=O(1)n1) with q* the canonical
inclusion. Here j*(w1) = 0 and j*(wi) = wi in H*(BSO(n)) for i > 1. Via i*,
we can identify H*(BO(1)n1) as P [x1, . .,.xn]=(oe1). The interpretation is t*
*hat
an nplane O(n)bundle , with a reduction of its structural group to SO(n) spli*
*ts
along q as the sum of n line bundles whose tensor product is the trivial line b*
*undle.
A NOTE ON THE SPLITTING PRINCIPLE 5
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397432.
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Mathematical Journal (2) 13(1961), 216240/
[4]R. Bott and H. Samelson. The cohomology ring of differentiable fiber bundles*
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[5]L.H. Hodgkin and V.P. Snaith. Topics in Ktheory. Springer Lecture Notes in *
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Department of Mathematics, The University of Chicago, Chicago, IL 60637
Email address: may@math.uchicago.edu