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 G-bundles 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 p-torsion. 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 p-subgroups 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 Bott-Samelson 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 R-module. Let EG be a universal principal G-bundle and take BT = EG=T and BG = EG=G. Inclusion of orbits gives a G-bundle 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 G-bundle 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 G-bundle or as a G-bundle with fiber F for any G-space 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 G-bundle 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 Eilenberg-Moore spectral sequence to obtain these conclusions. The map p is the universal example for the reduction of the structural group * *of a G-bundle from G to T . The G-bundle 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 G-bundle , over X, there is a G-bundle 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 n-circle 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 G-bundle 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 K-the* *ory, using its Serre or Eilenberg-Moore spectral sequence (see e.g. [5, 6]). In part* *icular, with this restriction on ss1(G), q* is a monomorphism in K-theory. 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 K-theory. 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 n-1 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 n-1_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 n-1) 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 n-plane 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 H-spaces and in particular to p-compact Lie groups, with the* *ir maximal tori. We leave such examples and elaborations to the interested reader. 4 J.P. MAY Using maximal 2-tori 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 n-plane 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)n-1 O(1)n be the subgroup of SO(n) O(n) consisting of the diagonal matrices of determinant 1. Let , be an oriented real n-plane bu* *ndle with classifying map f :X -! BSO(n) and define Y to be the pullback in the left square of the diagram g n-1 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)n-1) 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)n-1) as P [x1, . .,.xn]=(oe1). The interpretation is t* *hat an n-plane 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 References [1]A. Borel. Sur la cohomologie des espaces fibr'es principaux et des espaces h* *omog`enes de groupes de Lie compacts. Annals of Math. 57(1953), 115-207. [2]A. Borel. Topology of Lie groups and characteristic classes. Bull. Amer. Mat* *h. Soc. 61 (1955), 397-432. [3]A. Borel. Sous-groupes commutatifs et torsion des groupes de Lie compacts co* *nnexes. T^ohuku Mathematical Journal (2) 13(1961), 216-240/ [4]R. Bott and H. Samelson. The cohomology ring of differentiable fiber bundles* *. Comm. Math. Helv. 42(1955), 490-493. [5]L.H. Hodgkin and V.P. Snaith. Topics in K-theory. Springer Lecture Notes in * *Mathematics Vol 496. 1975. [6]V.P. Snaith. K-theory of homogeneous spaces and conjugate bundles of Lie gro* *ups. Proc. London Math. Soc. 22(1971), 562-584. Department of Mathematics, The University of Chicago, Chicago, IL 60637 E-mail address: may@math.uchicago.edu