Chapter 3 Equivariant Classifying Spaces and Characteristic Classes
3.3 Characteristic Classes for T 2 Vector Bundles
In this section, we focus on the special case G D T2, and we write A for a general closed subgroup of T2. We have two goals. The first is to give an equivariant form of the splitting principle, so that in Sec.4.2we can identify some characteristic classes of A-equivariant bundles over A-fixed spaces. The second is to record the calculation of the Borel Chern classes.
We write
log W A_ Š! H2.BAI Z/ for the first Chern class isomorphism between them.
We write z; z0for the generators of H2.BT2/Š Zz ˚ Zz0corresponding to the natural repre- sentation of T2, thus H.BT2I Z/ Š ZŒz; z0. We also use z; z0for their restriction to H2.BAI Z/, thus we can express H2.BAI Z/ as a quotient module
Zz˚ Zz0=K for some submodule K of
H2.BT2/.
Reductions and the splitting principle We describe the cohomology rings H.BU.n/A
/ and H.BS U.n/AI Q/ using the splitting principle. We start with U.n/: Since A is abelian, we may
choose our representatives m 2 Repn.A/ hom.A; U.n// of hom.A; U.n/ c
/=U.n/ to be of the form
mW A ! T; (3-13)
where T is the maximal torus of diagonal matrices. If m is such a homomorphism, then its centralizer
Z.m/D fg 2 U.n/jgmg 1 D mg Š Y
˛2A_
U.rank hom.˛; m//
is a product of unitary matrices. In particular, it is connected, with maximal torus T . We define W .m/ to be the Weyl group of Z.m/ with respect to the torus T ; it is a subgroup of the Weyl group W of T in U.n/. With these choices, Proposition3.1.1takes the following form.
Proposition 3.3.1 BU.n/A ' a m2Repn.A/ BZ.m/; HBU.n/A Š Y m2Repn.A/ H.BT /W .m/: (3-14)
Recall that in the isomorphism of A-bundles
V Š M
˛2A_
hom.˛; V / ˝ ˛;
A acts trivially on hom.˛; V /, while the bundle underlying ˛ is a topologically trivial line bundle. Thus if
hom.˛; V / Š L1˚ ˚ Ld
as a non-equivariant bundle, then
as a bundle with A-action. Proposition3.3.1implies the following form of the splitting principle.
Proposition 3.3.2 Let V be an A-equivariant vector bundle over an A-fixed space X. The splitting
principle holds in the sense that there is another A-fixed space X0and a cohomology monomorphism
X0 ! X so that over X0we may write
V Š M ˛2A_ d˛ M i D1 L˛;i˝ ˛;
where L˛;i is a line bundle with trivial action, and ˛ describes the A-action.
Let stand for U.n/ or S U.n/, and let T be a maximal torus, with Weyl group W . Suppose that
W P ! X
is an A-equivariant principal -bundle, over a trivial A- space X . The action of A on P corresponds to a section
s W X ! P hom.A; c/: giving a function
f W X ! P hom.A; c/ ! hom.A; c/=conjugacy:
Definition 3.3.3 A reduction of the action of A on the principal bundle P over X is a function
m W 0X ! hom.A; T /
making the diagram
X f- hom.A; c/= 0X ? m - hom.A; T / 6 (3-15)
commute. Note that a reduction always exists, because the right vertical arrow is a surjection of discrete spaces.
This definition is convenient for analyzing principal -bundles over not-necessarily connected spaces.
Let Z.m/ be the centralizer of m in . It is important to note the following.
Proposition 3.3.4 With D U.n/ or SU.n/, for any m W A ! T; the centralizer Z.m/ is connected, with maximal torus T .
Let W .m/ be the Weyl group of Z.m/; it is a subgroup of W . Any other reduction m0 W A ! T is of the form
m0D wm where w 2 W , and
wmD m
if and only if w 2 W .m/:
If X is connected, the reduction m determines a principal Z.m/-bundle Q.m/ over X , by the formula
Q.m/D fp 2 P js.p/ D .p; m/g:
This is classified by a map
gm W X ! BZ.m/:
By the splitting principle,
H.BZ.m/I Q/ Š H.BTI Q/W .m/;
The following proposition plays a central rule in the characterization of cohomology of fixed spaces.
Proposition 3.3.5 Let D U.n/ or SU.n/ as above. Let A be a closed subgroup of T . Then H.B AI Q/ is isomorphic to the ring
homW.hom.A; T /; H.BTI Q//
of W -equivariant functions. More explicitly, it consists of functions
W hom.A; T / ! H.BTI Q/ such that
(i) for each m 2 hom.A; T /, .m/ 2 H.BTI Q/W .m/; and
(ii) for w 2 W ,
.m/D w .wm/2 H.BTI Q/W .m/:
In particular, any such function determines a characteristic class of A-equivariant complex vector bundles over A-fixed spaces, by the formula
.V / D gm .m/;
where m W 0X ! hom.A; T / is any choice of reduction of the action of A on V =X. For D U.n/,
the analogous statements for integral cohomology are true as well.
Remark 3.3.6: The main ingredient in the argument is the splitting principle for BZ.m/; so one
needs to know that Z.m/ is a connected compact Lie group. Thus the result of Bott and Samelson [8] implies that the Proposition holds rationally for any simply-connected compact Lie group.
Remark 3.3.7: The results of this section and of Proposition3.1.1say that the components of B A are labelled by elements of
hom.A; c/= ;
where cdenotes as a -space with the conjugation action. A choice of representative m W A ! identifies the corresponding component with BZ.m/. One way to work with B A, then, is to fix a set of representatives. However, it is essential not what we will do, because we must understand the behavior of our characteristic classes under restriction
B T2 ! B A;
which leads us to consider diagrams like
T2 - T A 6 m - (3-16)
Our approach is to give formulas which work for any m W A ! T and which are compatible with the action of W by conjugation. Proposition3.3.5 tells us how to do this. When we write that a homomorphism m W A ! “labels a component of B A”, we mean that we use m to identify its component with BZ.m/.
Chern classes of T2-bundles Our calculation of Chern classes uses the splitting principle (Lemma
3.3.2) to deduce the general case from the following result, which is a specialization of Prop.3.2.1. Recall that there is an isomorphism log W A_ ! H2.BA/Š
Zz˚ Zz0=K, where K is the kernel of the surjection Z2 Š T2_ ! A_induced from the inclusion A ! T2. For ˛ 2 A_ we write log.˛/ D log1.˛/zC log2.˛/z0C K 2
Proposition 3.3.8 If L is a line bundle over an A-fixed space X D XA, and if ˛ 2 A_, then
c1T2.L˝ ˛/ D c1.L/C log1.˛/ z C log2.˛/ z0 2 HT22.X /
Now suppose that V is an A-equivariant vector bundle over an A-fixed space, and that after pulling back along a cohomology monomorphism X0 ! X we have
V Š L1˝ ˛1C C Ln˝ ˛n:
Then
cT2.V /DY
i
.1C c1.Li/C log1.˛i/zC log2.˛/z0/: (3-17) This gives a calculation of the first and second Borel Chern classes. In order to state the result, we introduce the following quantities. Suppose that m D .m1; : : : ; md/ and m0 D .m01; : : : ; m0d/ are
arrays of elements of Z. Let
.m/W D X i <j mimj I.m; m0/W D X i ¤j mimj0:
Similarly, if .x1; : : : ; xd/ are elements of a Z-module X, then
I.m; x/W D X
i ¤j
mixj:
We have chosen the signs of and I so that the right hand sides appear with positive sign in the following.
Proposition 3.3.9 (i) is quadratic, I is symmetric and bilinear, and (a) .km/ D k2 .m/
(b) .m C m0/D .m/ C I.m; m0/C .m0/ and hence 2 .m/D I.m; m/
(c) I.km; m0/D I.m; km0/D kI.m; m0/
(ii) If P mi D 0, then
I.m; x/DX
i
mixi:
(iii) If P mi D 0 then, then
2 .m/DX
i
m2i:
Proposition 3.3.10 Writing mi D log1.˛i/; m0i D log2.˛i/ and xi D c1.Li/, we have
c1T2.V /D c1.V /C . X i mi/ z C . X i m0i/ z0
c2T2.V /D c2.V / I.m; x/z I.m0; x/z0 .m/z2 I.m; m0/zz0 .m0/z2:
(3-18)
The Borel equivariant sigma orientation Recall that we defined the sigma orientation in Eg.2.4.5, in this paragraph we write down the T2equivariant Euler class of sigma orientation.
Suppose that V is an T2- equivariant S U.d /-bundle over an T2-fixed space, given as
V Š L1˝ ˛1˚ ˚ Ld ˝ ˛d; (3-19)
where Li is a complex line bundle with Chern class xi and ˛i 2 .T2/_: Let mi D log1˛i; m0i D
log2˛i 2 Z. Then the equivariant Euler class of the sigma orientation is
eT2.V; /D
Y
i