Proceedings of the International Congress of Mathematicians
Vancouver, 1974
Invariants of Flat Bundles
Jeff Cheeger
Let G be a Lie group with finitely many components. We are going to discuss
some cohomology invariants of flat principal (/-bundles. Their properties were
developed jointly with Simons [2] in an outgrowth of earlier work of Chern and
Simons [3]; see also [4] for related ideas.
We begin with some notation. Let /*(<?) denote the ring of invariant polynomials
of G and w: Ik(G) -> HU(BG> R) the universal Weil homomorphism. Fix A<£.R,
a proper subring and define
K*(G9 A) = {(P, u) E P(G) x H*KBG> A) \ w(P) = r(u)}.
Here r(u) e HU(BG, R) is the real image of u. Let G° represent G equipped with
the discrete topology. BGo = K(G°, 1) is the classifying space for flat principal
G-bundles. Let b: H2k^( , R/A) -> #2*( , A) denote the Bockstein map of the
coefficient sequence 0 -> A -> R -> R/A -> 0.
1. There is a homomorphism S; Kk(G, A) -+ //2*-1(i?Go, R/A) such that
the following diagram commutes
THEOREM
K*(G, A)
H^KBG*> R/A) zt HHBG°, A) £. H**(BG9 A)
Equivalently, for each flat bundle a over a space M there exists SPtU(a)e
H2tt~l(M9 R/A) which is natural under bundle maps such that b{SPtU(oc)) =
- w(a)ei/2*-i(M).
The Sp)U are secondary characteristic classes in the sense that unless P E O ,
they are not the pull-back of classes in BG,
©1975, Canadian Mathematical Congress
4
JEFF CHEEGER
For the details of their construction and its extension to the more general context
of nonflat bundles with connection we refer to [3].
K*(G, A) = ®Kk (G, A) forms a graded ring with multiplication (Pis U{)
.(p 2s U2) = (Pi-Pto Ui U Uà- H*(R/A) is also a graded ring with respect
to the »-product fx*f2 = (-1) kib *f) U h where f{ e H"( , R/A). Note that
if A = Q, the rationals, then b = Q and the »-product is trivial. Also it may be
shown that S is a ring homomorphism.
In case the holonomy group H of a is finite then, as J. Millson pointed out, the
SP,U have an explicit interpretation. Let p: BH -» BG. Since H is finite, H*(BH, R)
= 0 for i ^ 1 and b : H2i~l(BH9 R/A) -> H2i(BH, A) is an isomorphism. Then
necessarily we must have H2i~l(BH9 R/A) 3 SPiU = - b(p*(u)). In case H is infinite, it is no longer true that H^Bx, R) = 0, but the formula may still be generalized by introducing so-called Borei cohomology.
It is also possible to construct the SPiU intrinsically:
EXAMPLE 2. Let G = SO(2«), # be the Euler class and Px the Pfaffian. Let a be
a flat (/-bundle and S(E) the associated sphere bundle. Then we have the homology sequence
0 -> #2»-i(s2«-i) _> H2n(S(E)) -> H2»(M) -> 0.
I*
Z
Let Vol denote the volume form on the fibre normalized so that Vol (S2n~x) = 1.
Since S(E) is flat, Vol extends uniquely to a closed form o> on S(E). Given x e
H2n-i(M9 Z) choose z e Z2w_i(J5'(M)) such that |>r(z)] = w. Set SPxtX(x) = %(x)
= (Cü(Z)) ~ where ~ denotes reduction mod Z. It follows from the exact sequence
that x is well defined and one may show that % = SPxt%.
In analogy with Example 2 it is possible to represent % by an explicit Borei cocycle in the bar resolution of G°. The formula for this cocycle involves the transcendental function which expresses the volume of a totally geodesic simplex on S2""1
in terms of its dihedral angles. We mention in passing that by means of the power
series for this function it is possible to construct a/?-adic analogue of %.
With appropriate modifications, the discussion of % applies equally well to classes
ci9 pi corresponding to Chern and Pontryagin classes of GL(«, C) and GL(«, R)
bundles. Set c = 1 4- Tn>o à{. if we regard c as being defined for flat vector bundles
then the expected relation c(E ® F) = c{E) * c{F) holds. Moreover, taking A =
Q we have (ch) corresponding to the Chern character. Because the *-product
is trivial mod Q,
(3)
(&) = » + Z J ( - l ) ' - i ,. * ' n ,
and
(4)
(elk) (En* ® E*) = n2(ch) (EnC) + n^Bi) (En*) - nxn2.
An important property of the SPiU is their rigidity under deformations in
dimension > 1.
INVARIANTS OF FLAT BUNDLES
5
THEOREM 5. Let 6t be a X-parameter family of flat connections on the bundle
G -> E -> M, Let SPìU e H2kr~\M9 R/A) be an invariant corresponding to 0t. If
k > l9SPitl is independent oft.
This result is somewhat mysterious in that the proof makes use of nonflat
bundles. Observe that for flat circle bundles over the circle, % e H^S1, R/Z) is
just given by the angle of holonomy which can take on any value. It is therefore
not rigid.
The deepest questions about the SPtU relate to the values which they take on
cycles. Let V denote the vector space over Q generated by the values of SPtU e
H2k'\BGo9 RIZ). Although for k > 1 the SP>U are not known to take on irrational
values, it seems reasonable to conjecture that the dimension of V is countably
infinite. In fact if G is algebraic, an argument based on the rigidity theorem shows
that the set of values of SPtU is countable for k > 1. The dimension of V furnishes
a lower bound for the rank of the homology group H2k-i(BGo9 R). However, these
homology groups can often be shown to be infinite by more indirect arguments.
For example,
THEOREM 6. The groups H2n-i($0(2n, C)) and H2n-i(SO(2n, R)) and
H2n-i(SO(2n — 1,1, R)) all have infinite rank,
The proof uses what might be called "real noncontinuous" cohomology classes
to show that certain cycles constructed from number theory are independent. In
fact, the invariant polynomial P% on SO(2w, R) extends by complex linearity to
an invariant polynomial on SO(2w, C). If IP% denotes the imaginary part of this
polynomial, then w(IPx) = 0 and we have Jx = SIPx,0 e i/r2w_1(^so(2«,c)» ^)« Let
a be an automorphism of the algebraic numbers. Then G may be extended to a
(noncontinuous) automorphism of C and hence induces an automorphism of a* of
jj2n-i (BSO(2ttiC)9 R). The relevant classes are of the form G*(JX) for suitable a.
We should not neglect to mention the important problem offindinga natural and
complete set of invariants for deciding when a cycle in H%(BGo9 Z) is homologous
to zero.
We will close by describing how the SPfU arise in the context of the "geometric
index theorem" of Atiyah, Patodi, Singer [1]. They have defined a spectral invariant
7]& (M2Ä_1, g) of a flat unitary bundle En over a riemannian manifold M2k~l
with metric g. If E1 is globally flat, we just write 7](M2k~l9 g), k odd implies
2
= 0.
v(M *-\g)
Now assume that M2k~l = dN2k and that En extends to a possibly nonflat
bundle Fn over N2*. Choose a metric g on N2k such that on U = dN2k x [0, e] c
N2k9 g is isometrically the product of g and the metric on [0, e). Likewise, let v
be a connection on Fn\ U which is the product of theflatconnection on En and the
trivial connection along [0, e]. Let Q9 0 denote the curvature forms of g, v and
L9 ch denote the Hirzebruch and Chern character polynomials. Then, there is an
integer # which is the index of a certain boundary value problem associated to
the data such that
6
JEFF CHEEGER
- f = Jjp L(fl) A <4(0) + ( - 1 ) *
VE(g).
We observe
LEMMA
7. j ^ , iL(fl) A ch(@) = L(M) U (ch)(En)(M) mod ß , so fAûf
( - 1)*+1 9£Cf) = i ( A O U (ch)(E»)(M)
mod ß .
The proof while not standard is quite short. Combining Lemma 7 with (3) and
(4) gives
( - l)k+lyE*(M,g)
(8)
= (-
\)k^n^(M9g)
+ | { ^ r i ) l L*M U ^^(Ä-XAO mod ß
and
(9)
^«,®i?»2 W #) = «2%», (AG #) + «î^», (M, g) - Wi«2^ (M, g) mod ß .
Because these formulas only hold mod Q, it seems doubtful that they can be
proved directly by purely analytic methods.
Bibliography
1. M. F. Atiyah, V. Patodi and I. M. Singer, Spectral asymmetry and riemannian geometry
(preprint).
2. J. Cheeger and J. Simons, Differential characters and geometric invariants (preprint).
3. S.S. Chern and J. Simons, Characteristic forms and geometric invariants, Ann. of Math. (2)
99 (1974), 48-69.
4. F.W. Kamber and P. Tondeur, Characteristic invariants of foliated bundles (preprint).
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