Izv. Akad. Nauk S S S R
Ser Mat. Tom 38 (1974), No. 6
M ath. US S R I z ve s dja
Vol. 8 (1974), No. 6
FORMAL GROUPS AND THE ATIYAH HIRZEBRUCH FORMULA
UDC 513.83
I. M. KRICEVER
Abstract. In this article, manifolds with actions of compact Lie groups are con
sidered. For each rational Hirzebruch genus A: Q* —• Q, an "equivariant genus" h ,
a homomorphism from the bordism ring of G manifolds to the ring K(BG) ® Q, is con
structed. With the aid of the language of formal groups, for some genera it is proved
that for a connected compact Lie group G, the image of h belongs to the subring
Q C K(BG) ® Q. As a consequence, extremely simple relations between the values of
these genera on bordism classes of S manifolds and submanifolds of its fixed points
are found. In particular, a new proof of the Atiyah Hirzebruch formula is obtained.
Bibliography: 10 items.
In [ l ] it was proved that the signature of every S
of the signatures of the submanifolds Fs
manifold X i s equal to the sum
of its fixed points:
s
As a unitary variant of this formula, there is a relation between the values of the
c las s ic al Ty genus of an S ^manifold, i. e. an almost complex manifold on which the
action of S
preserves the complex structure in the stable tangent bundle, and it s fixed
submanifolds:
Here e~ i s the number of summands in the decomposition of the representation of S 1 in
the fibers of the normal bundle over the submanifold Fs
into irreducible representations
η'*' ( the action of ζ e S 1 in 77* i s multiplication by 2*) with j
s j
< 0. We shall denote
by ( g the number of the remaining summands.
The previous proof of both formulas in [ l ] is based on the index theorem of Atiyah
and S inger. In this article they are obtained as a consequence of a fundamentally dif
ferent approach, whose e s s e nc e is reduced to the study of the analytic properties of the
Conner Floyd expressions.
We recall these expressions for actions with isolated fixed points ( s e e [ 2l, [ 3l and
AMS (MOS) subject classifications
57D15, 57D90.
(1970). Primary 57A65, 53C1O, 53C15; Secondary 55B2O,
Copyright S> 1976. American Mathematical Society
1272
I. M. KRICEVER
[4]; for an arbitrary action they were first obtained in I5J, whose formulas were improved
in [61).
Assume that the action of 5
on X h as only isolated fixed poin ts p
and that
the represen tation of the group in the fibers of the tangent bundle over them is
Σ"
T? 7 S I
If [u\ .
i s t he
;t h
p o we r o f
u in t h e fo rma l g ro up o f "g e o m e t r i c
c o bo rdi s ms "
oo
l
/ («, σ)
g~ (g (u) + g (υ)), g (u) = 2
π—ο
i.e. luI · = g~ (jg(u^), then the equalities of Conner and Floyd assert that the Laurent
series
with c o effic ien t s in U ® Q, c o n t a in s on ly t h e righ t p a r t , an d t h e in d ep en d en t term i s
t h e bordism c l a s s of X.
T o eac h r a t io n al H irzebru ch ge n u s h: U —• Q t h e r e c o r r esp o n d s a n u m erical real
iza t io n Φ(ι/ ), a se r i e s in 1
1
where g j ^ ^
η with r a t io n a l c o e ffic ie n t s:
is functionally in verse to the logarithm
of th e formal group fSu,
v\ c o r r esp o n d in g to t h e hom om orphism h. We n o t e t h at
Φ ^ ( ^ i s t h e im age of t h e se r i e s Φ ( «) , to which t h er e c o r r e sp o n d s a cobordism c l a s s
in U*(CP°°) ® Q = U*[[u]] ® Q, un der t h e hom om orphism £ : U*(CP°°) ® Q —
00
KiCP )
® Ο in d u ced by t h e ge n u s / '. T h e e xi st e n c e of t h e tran sform atio n of fun ctors
/> follows from [7].
We sh a ll a ssu m e t h a t t h e se r i e s gT
(in rj) i s t h e exp a n sio n about 1 of an a n a lyt ic
fun ction in som e n eigh borh ood of t h a t p o in t ; t h en from th e C o n n er F lo yd e q u a lit ie s it
follows t h a t Φΐ,( 7?) is a n a lyt ic in som e n eigh bo rh o o d of 1, and t h a t Φ ι / ΐ ) = M [Xl)·
Our t a sk is to prove t h at if th e fun ction g~
(In η) is a n a lyt ic in t h e d isk |τ;Ι < 2
an d d o e s n ot h a ve ze r o s t h e r e , exc ep t at 1, t h en Φ Λ(»?\ wh ich could h ave p o le s at t h e
r o o t s of 1, is a lso a n a lyt ic in t h e d isk \ η\ < 2. F rom t h is it follows t h a t , for a ge n u s
h such t h a t g^ ' ( I n η) i s a r a t io n a l fun ction with on e sim p le zero at 1, i. e . gj~ '(ln η) =
(η
ΐ)/ (αη + b), a + b = 1, Φ / , (Ϊ?) i s a n a lyt ic everywh ere and h e n c e is a c o n st a n t . T h en
i t s valu e at 1, equ al to / / [ Xl\ c o in c id e s w> h
£
t
e
7
1ϊΐηΦ Λ(η) = y\ a * (~b) \
η—=°
s
P r e c i se ly in t h i s way, a n ew proof of t h e At iyah H irzebru ch form ula will be ob
t a in ed ; for t h e two param eter ge n u s Τ
( t h e va lu e of Τ
on t h e bo rdism c l a s s
FORMAL G ROUPS AND TH E ATIYAII HIRZEBRUCH FORMULA
n
η
[CP ]
i s equal to Σ"_ οχ ~
''
yV; we n o t e t h a t T j
1273
c o in c id e s with t h e Τ
gen u s)
we sh a ll o bt ain t h e relat io n
s
I t a ke t h is o c c a sio n to e xp r e ss d e e p ackn o wled gem en t to S. P . N o viko v, V. M.
Bu h st a be r an d S. B. Slosm an for t h e ir in t e r e st in t h is work and va lu a ble a d vi c e .
§ 1 . "C h a r a c t e r i s t i c " h om om orph ism s for G bun d i e s
T o eac h c h a r a c t e r ist ic c l a s s \
6 Ul(BU) of com plex vec t o r bu n d le s in u n it ar y
cobordism t h er e c o r r esp o n d s a " c h a r a c t e r i s t i c " hom om orphism
U "+i(pt),
χ · : Un (BU(k))—• *£/ + <=
wh ich a sso c i a t e s to a bun dle ξ over t h e m an ifold X t h e im age of t h e cobordism c l a s s
χ(0
u n der t h e c o m p o sit e
If (Χ) Ζ Un i (X)
£/„_,· s iT"+ '.
where D is the duality homomorphism.
In this section, an analogous homomorphism for complex G bundles (here and in
what follows G is a compact Lie group) will be constructed and studied.
1. Consider the category of complex (/ bundles over unitary G manifolds. Two
su c h bu n d le s ζι
an d ζ2
o ver G m an ifolds <Yj and X2
G bun dle ζ o ve r W su c h t h a t dW = X.U
are bo rd an t if t h e r e e xi s t s a
X~ an d t h e r e st r ic t io n of ζ to X., i = 1, 2,
G
c o in c id e s with ξ . T h e bordism grou ps o bt a in e d in t h i s way will be d en o t e d by U
wh ere k
d i n v ii an d η
with Un(BU(k)).
su bm o d u le I/ ,
T h en (jf
0
e
d im _ X. In t h e c a s e when G = \ e\\ i s t r ivia l, U
, co in cid es
„ be c o m e s a rin g an d a l^ m o d u le in t h e u su a l way.
i s id en t ified with t h e bordism m odule U^
We sh a ll d e n o t e by Xc
The
of u n it ary G m an ifolds.
t h e sp a c e (X x EC)/ C, and by £
c
t h e im age of t h e G
bu n dle ζ u n der t h e hom om orphism
Ve c t e ( *) —
If p,: U (Xc)
p: X^
—· U (BG)
i s t h e G ysin hom om orphism in d u c ed by t h e p r o jec t io n
—• BG, then t h e form ula
χο(1ί1)=ρ·(χ(1ο))
defines an "equivariant characteristic" homomorphism
I t s relat io n with y
is given by t h e followin g lem m a.
Lem m a 1.1. Let L·' , —· Ue , be the homomorphism
action;
then the
diagram
which "forgets"
,,
the G
1274
is
I. M. KRICEVER
commutative.
The proof of the lemma follows immediately from the definition of the Gysin
homomorphism and from the fact that the restriction to the fiber X of the fibering
Xr —• BG of £ ._ coincides with f.
In what follows, it will be convenient to denote the "characteristic" homomorphism
1
G
corresponding to 1 eU°{BU)
by
2. Let Η be a normal subgroup of G. The set of fixed points under the action of
Η on a unitary G manifold X i s the disjoint union of almost complex submanifolds
F
( not nec essarily connected). The normal bundles vg
over the submanifolds
F
have a natural complex G bundle structure.
As is known, there e x is ts an equivariant embedding of X into the space of a uni
tary representation ∆ of the group G. ( To avoid repeating conditions, we shall agree
that in this section and in the next one, only unitary manifolds, bundles, representations,
etc. , will be considered. ) D enote the restriction of the normal G bundle over X in the
space of the representation ∆ to the submanifold Fs by ( ί ζ ) . It is evident that the
sum vs
θ (
D^) i s a trivial G bundle. Let ∆ be a maximal direct summand of ∆
whose restriction to the subgroup Η does not contain trivial representations of H. In
an analogous way we s e le c t a direct summand (
Theorem 1.1. Let ξχ
X to a submanifold
be the restriction
vg)
in the G bundle (
v^).
of a complex G'bundle ζ over a G manifold
F . Then, with the notation introduced
s
above,
Χΰ(\ 1)) = 2 P« («(( vs)c)
s
where ps,: U (F c) —> U {BG) is the Gysin homomorphism and, for an arbitra
bundle ζ, ε{ζ) is the Euler class of ζ.
Proof. The composite of the embedding of X in the s pac e of the representation
A and the projection on the direct summand ∆ defines an equivariant map h: X —» ∆ .
For each bundle ζ we shall denote by Εζ it s total spac e and by Ξζ its sphere
bundle. ( ∆ x EG)/ G coincides with EAG by definition. Here and in what follows,
∆ denotes both the representation and it s space.
Let h x id: X x EG —· ∆ x EG; then the corresponding quotient map
induces a Gysin homomorphism
A,: ί/ ·(Χ0 )^ί/ ·(£ ∆ β , SA.).
Lemma 1.2. Let *: l/ *(EAc, S\ C) —* U'iBG^ be the homomorphism from the
FORMAL GROUPS AND THE ATIYAH HIRZEBRUCH FORMULA
1275
exact sequence of the pair; then for every χ e U (XQ) the following equality is sat
isfied:
Proof. From the definition of the G ysin homomorphism it follows immediately that
h,(x) =
t(AG)pdx),
where t(Ac) is the Thom class of ∆ ^ . Apply to both sides of th is equality the homo
morphism i . The lemma follows from the fact that i t(Ac) = e(Ac).
We shall st ate a simple corollary of Lemma 1.2. We denote by / *(G ) the ideal of
U (BG) consisting of those cobordism classes which are annihilated by multiplication
by the Euler c la sse s of bundles associated with represen tation s of G.
Corollary. / / the action of the group G on the manifold X has no fixed points,
then the image of p,: U*(XC) —» U*(BG) belongs to the ideal I*(G).
Proof. If Η coin cides with G, then, by definition of h, the nonexistence of fixed
poin ts implies that the image of XG belongs to SAG· T h is means that i*h, is a trivial
homomorphism. By Lemma 1.2, the image of />, is annihilated by multiplication by
e(AG).
If we return to the proof of the theorem, we note that for an arbitrary action of G
on X, h maps the pair (XG, N G) to the pair (EAG, SAG). H ere Ν is the complement
of tubular neighborhoods of the fixed poin ts under the action of H. The restriction of
A to a closed tubular neighborhood of F^ defines a map of pairs h : (EusG, S »/ j C ) ·
(EAG, SAG) which induces a G ysin homomorphism:
hs] : IT ( £ v s 0 ) = U* (Fs0)
Lemma 1.3. / / / : Ev
. —· Xr
> U* (BG).
is the inclusion, then
hs\ °fi'.U* (Xc) —*• U* (BG)
and
^
^s! ° A
=
l
* ° hi
s
Proof. The G ysin homomorphisms induced by the maps of the commutative diagram
EAa ^(EAG,
Xa
»(XG,N C)
also form a commutative diagram:
U'IEAG,
WI
U'(Xa)
The natural identification of XG\ N G
the lemma.
SAQ)
ί
U'(Xa\ N
with the disjoint union of the EvsG
Lemma L4. For χ e U*(Fs(.) the following equality holds:
will yield
1276
I. M. KRICEVER
Proof. The map hs can be factored as the composite
( £ v s C , Svsa)
— (EAc,
Stic)
where g i s the quotient map of the equivariant map
£ vs xEG *Fs
obtained from the projection Eus —> Fs,
χ ∆ χ EG,
the equivariant inclusion of Evs
the identity map of EG. This means that hs,(x)
= ps,g,(x)·
in ∆ and
The map g is the identity on
the base of the bundles; therefore g*(x) = χ and g,(x) = g, ( g*( x)) = xg, ( l). It remains to
show that g, ( l) = e ( (
vs) c).
According to the definition of h, g i s the embedding with normal bundle (
However, for any bundles £ j and ζ2
^s)
c
over the common base, the "diagonal" section of
τ*ζ2 ( τ : Ε(ζι Φ ζ2) —» Υ i s the projection on the base) i s transversal to the zero
section and their intersection is the image of the embedding i: (Εζ^, S £ j) —•
( E( £ j Θ £ 2 ) , S ( £ j Θ 4^2^· F r o m
t n e
definition of the Euler class and of die homomorphism
i, it follows that i, = e ( £ 2 ), which concludes the proof of the lemma.
The theorem follows immediately from the preceding lemmas and from the fact that
Remark. In the c as e when the subgroup Η c oinc ide s with the group G, Theorem 1.1
give s a relation between the value of the homomorphism χ
on the bordism c la s s of
the G bundle ζ and invariants in the cobordisms of the fixed submanifolds.
3. Along with the expression for y .
given by Theorem 1.1, in what follows we
shall need a modification of it, which will be obtained pre c ise ly in this subsection.
Let F
be a connected component of the set of fixed points under the action of
S 1 on an S ' manifold X. The normal bundle νs, like every complex S ' bundle over a
trivial 5^manifold, will be represented in the form Σ./ Οι^
Ι
0 η', where η' i s the / th
Γ
Ί
tensor power of the standard representation of S η, as in the Introduction ( s e e l8J).
The collection of complex bundles νχ., of which only a finite number are different
from zero, defines a bordism c la s s belonging to the group
The summation i s taken over all c olle c tions of nonnegative integers n. and / such
that 2Ση. + / = n.
The sum over all the connected components of the s e c l a s s e s give s the image of
the bordism c la s s of the S
manifold X, [X, S ] e U
, under the homomorphism β:
We choose as generators of the t/ ^ module Ut(CP°°)
n
00
c l a s s e s (CP ) e U^CP )
= ί/ φ (ΒΙ/ (ΐ)) the bordism
corresponding to the inclusion of CP" in CP™ or, what
is the same, the canonical bundle η. . over CP". The standard multiplication in Rt
allows us to choose as generators of the 1/ ,, module R t , in this c as e , the monomials
χ
...
FORMAL G ROUPS AND THE ATIYAH HIRZEBRUCH FORMULA
1277
It will be c o n ve n ie n t to d en o t e by η n ot on ly t h e c a n o n ic a l r e p r e se n t a t io n of S
but a lso t h e c o rresp o n d in g c a n o n ic a l bun dle over CP°°; t h a t i s, η . = η. T h en for t h e
S' bundle
n
η(η)
f]( n) ® ψ T ^
o ver CP ,
e
com plem en tary to η/ ηγ
wh ere f(u,
t h e bu n dle (η(η)
E u le r c l a ss of (
® η)
l
over CP" χ C P °° i s equ al to
η, η<) ® η, wh ere ( 7?/ n ) ) i s t h e « d im en sio n al bu n d le
i s defin ed by
1
ν) = Λη ® η) = u + ν + —CL.U'I· ,
th e formal gro u p of "geo m et r ic " cobordisms.
H e n c e , if
An(u,
^ ^ ( ( ^ n J ^ r O e E t r i C P x C / ' ^ t / ' f t u , v]]/ v«+<=0,
then
An{U,V)=
y ϊ
.
u '
L e t Bn(u)
be t h e im age of An(u,
v
'
CP00)
v) un der t h e G ysin h om om orph ism U"(CP" χ
—*U (CP°°) in d u c ed by t h e p r o je c t io n . We n o t e t h at for Aj<u, v) t h is bom om orph ism
k
c o r r esp o n d s to t h e su b st it u t io n of [CP"~ ]
for
iA.
Th eorem 1.1 im m ed iat ely yie ld s t h e followin g a sse r t i o n .
Theorem 1.2. There
exists
such that Ψ ο β coincides
U [[uW ® Q[u~
a U^ module homomorphism
with the composition
]. The values
of \
of Ψ on the generators
Ψ: R + —> £/ *[[«]] ® Q[u~
and the inclusion
Q
of the U
U
(CP°°)—·
module are given
by the
formula
/
r
ι
r
\
\ m= i
n
A. C o n sid er an arbit rary hom om orphism a : G j —· C of L i e gr o u p s. I t i n d u c e s a
m ap a *: / JCj —> BG of u n ive r sa l c la ssifyin g sp a c e s an d h e n c e a h om om orph ism
a*: U*(BC)~*
U*(BGl).
On t h e o t h e r h a n d , by m e a n s of α eac h C bun dle be c o m e s a G j bu n d le, i . e . t h er e
e xi st s a hom om orphism
T h e com m u t at ive diagram
Xc,
*• Xo
BGBG
wh ere .V i s an arbit rary G manifold, e a sily yie ld s
Theorem 1.3· For every characteristic
is
commutative.
class
y , the
diagram
1278
I. M. KRICEVER
§2. Equivariant H irzebruch genera. Statement and proof
of the main theorem
1. From the viewpoint of ch aracteristic classes, a rational H irzebruch genus, i.e.
a homomorphism h: U + —• Q, is given by a series t/ h(i) with hit) = t + Σ . > ] λ ι . ί', λ^ e Q.
The "a c t io n " of such a series on the bordism c lass [ C P n ] is given by the formula
where [r(u)] denotes the nth coefficient of the series r(«).I n [2], S. P. Novikov proved
that Mi) coincides with the series gj^ ' ) functionally inverse to the logarithm
n—0
of the formal group ftXu, v) which is the image of the formal group of "geo m et ric"
cobordisms under the homomorphism.
By a theorem of Dold [7], to each rational H irzebruch gen us h there corresponds
a transformation of functors 7>: U*(Y) —» Κ**(γ) ® Q (where K** is the Z 2 graded K
functor) such that h: U* —* Q coincides with the composite I/ * ^ U % £• Q.
The proof of the following lemma is analogous to the proofs of Theorem 6.4 and
Corollary 6.5 in [9Ϊ:
Lemma 2.1. The value of the homomorphism h at the generator u € U2(CP°°) is
equal to ch~ HgJ~ l(t)), where ch is the Chem character; that is,
h («) = gn1 (In η) s Κ (CP°°) ® Q = Q [[1 _ η]].
Um
Definition. An equivariant Hirzebruch genus corresponding to a rational genus h:
c
» Q is a homomorphism d = L x J : l ^ v » K(BG) ® Q.
Since Lemma 1.1 implies the commutativity of the diagram
I
I
i
where f: K*(Y) ® Q —* Q is the "augm en tation ", we have
Lemma 2.2. The value of a genus on the bordism class of a G manijold X is equal
to t(Ac ([X, G})).
2. Now we proceed to prove the main result.
Theorem 2.1. For a connected compact Lie group G, the image of the homomorphism
1
C C
T G y :U
: U
ev ev —» KiBG) ® Q belongs to the subring Q C K(BG) ® Q. Moreover, for an S
manifold X,
The H irzebruch genus Τχ
and the nonnegative in tegers f* and <~ appearing in
the statement are the same as in the Introduction.
FORMAL GROUPS AND THE ATIYAH HIRZEBRUCH FORMULA
1279
Proof. First of all we shall show that the first part of the theorem (im Τχ
C Q)
i s a simple consequence of Theorem 1.3 and Lemma 2. 3.
Lemma 2 3. The image of T^
1
belongs
to Q C K(CP°°) ® Q.
Indeed, for a connected compact Lie group G the homomorphism α : K(BG) ® Q—*
K(BH) ® Q induced by the inclusion of a maximal torus Η in G i s a monomorphism.
Therefore, if there e x is t s a G manifold X such that T^
a*{T^y([X,
S
1
G])) 4QQ
{[X, G]) 4 Q, then also
K(BH) ® Q. Evidently, there is an embedding of S
1
in Η, α ^
—» H, such that a *( a*( T^ y ( [ X, G] ))) doe s not belong to Q either. However, this
contradicts Lemma 2 . 3 be c ause by Theorem 1.3
αϊ (α*( J t , ([X, G]))) = 7 j , ; ([X, S 1]).
Proof of Lemma 2.3. Consider an S ' manifold X. Let
I
m
then by Theorem 1.2
7?*0*. S1])
Σ 7V,([.W,]
We shall calculate Τχ y(lu].)
and Τχ y(BNiu)).
S ince
we have
( 0
|
r
1
π
Therefore
and hence
Z (/ In η)
By definition of BN(u),
efficients of the s e rie s AN(u,
to find Τχ
(BN(u))
v) and replace vk
we have to apply Τχ
by Τχ
{[CP N~ k])
s e rie s ANT
(u, v). S ince
" * x,y
we have
ANTXty ( «. V) Ξ
Therefore
" U+ y*™)
( m o d WAr+i)_
to the co
in the resulting
1280
I. M. KRICEVER
ANTxty (u, υ) = 2 (
l)Vu" * (1 + (y
x) u)
k
i0
+ (i/ — x) uo) k .
T h u s we o bt a in
• ?\ _ ι ,*r«
We d e n o t e by Γχ
{η) t h e fun ction of η given by t h e righ t sid e of t h is eq u a lit y.
With t h e p r e c e d in g form ulas, th e eq u a lit y ( 1 ) t a ke s t h e form
m \
i
η
— y
Le t us pa u s e to c o ns i de r in de t a i l the me aning of the lat t e r e qua lit y.
L e t Φχ
{η) b e t h e fun ction of t h e com plex va r ia ble η given by t h e righ t sid e of
( 2) . I t is e a sy to se e t h at in a d e le t e d n eigh borh oo d of 1 it i s a n a lyt ic ; h e n c e it h a s
a L a u r e n t se r i e s exp a n sio n in t h e va r ia ble
with Τχ
([X, 5 ]) e Q[[l
1 — η t h e r e . By ( 2) , t h is se r i e s c o in c id e s
η] ] . T h i s im p lies t h a t Φχ
(η) i s a n a lyt ic not on ly in a
d e le t e d n eigh bo rh o o d , but a l so a t 1 it se lf.
Our im m ed ia t e t a sk will be t o p ro ve t h a t t h er e a r e n o p o le s at r o o t s of 1 an d, a s
a c o n se q u e n c e , t h a t Φχ
Lemma 2. 4. Lei
(η) i s a n a lyt ic in t h e wh o le p la n e .
χ = x/ (x + y) and y = y/ ( x + y).
Then
T h e proof of t h e lemma can e a si ly be o bt ain ed from t h e fact t h a t
Xf ([X, &})
S
U~*N ICP"),
f ~ ~ (u) =
x y
'
*=±
=
xi\ +y
(x + y)
By t h i s lem m a it is en ough t o c o n sid e r t h e c a se when χ + y •
xr\
1, wh at will be
assu m ed t ill t h e c o n c lu sio n of Lem m a 2. 3.
Assu m e t h a t Η ( t h e n orm al su bgro u p a p p ea r in g in § 1.2) i s a c yc li c subgrou p of
S
of o rder n. In t h e n o t at io n of T h eorem 1.1,
1
«(As·) Xf (\ X, S ]) = 2 ps! ( e (
s
Sin ce by d efin it io n of t h e r e p r e se n t a t io n ∆ of S
i t s r e st r ic t io n to t h e su bgro u p 7
d o e s n ot c o n t a in t rivial su m m an d s, we h a ve ∆ = Σ ^ η m,
vi si b le by n. H e n c e
vs)s·).
wh ere n o n e of t h e j
m
is di
n
FORMAL GROUPS AND THE ATIYAH HIRZEBRUCH FORMULA
1281
[Π ( η ' *~ 1 ) 1 Tf, y ([X, S>]) = S T x.y [ft, (e ( v.)s.)].
N ow we c o n sid e r an arbitrary S
bun dle ζ over an S ' m an ifold F su ch t h a t t h e
a c t io n of Z n i s t r ivia l on F. ζ c a n be r e p r e se n t e d a s a sum of 5
r <η
1. T h e ge n e r a t o r of Ζ
exp {2πίτ/ η).
H e n c e , if t h e S
(3)
a c t s on a fiber of ζ
bun dle ζ
bu n d les ζτ, Ο<
by m u lt ip lic a t io n by
i s £ 0 7 / " ' , t h en Ζ
a c t s t r ivia lly on ζ .
Sin ce £ r = ζτ ® η Γ , we h a ve
wh ere p: F j • C P °°.
^
*\,
L e t µ^ , be t h e Wu ge n e r a t o r s of ζ
j . T h en
^.y (P. (e (Cs·))) = 'A .y [pi ( Π/ 0*r.*. P* (I«
T h e co efficien t of p*([u] Υ in t h e se r i e s
i s a sym m etric p o lyn o m ia l in µτ ,. We d e n o t e t h e c o r r esp o n d in g p o lyn o m ia l in t h e
C h ern c l a s s e s of ζ
. by P. ,. T h e d im en sio n of i t s lo we st term i s n o t sm a ller t h a n
i —dim ζ .
Thus
f r) f{ >(f[Pir.r))>
2 f ( i l
T h e p r o jec t io n a : S
» S / Zn = S
of 5
« = (iv . . . , »„ ,).
(4)
on to t h e qu o t ien t group in d u c e s a m ap
of c la ssifyin g sp a c e s
a , : CP°° » CP°°,
with a (u) = [u\ n· Sin c e t h e S
S
ζτ),
bun dle ζ
bun dle ζτ
i s t h e in ve r se im a ge u n d er α
( we r e c a ll t h a t t h e su bgro u p Ζ
P
it follows from T h eorem 1.3 t h a t pJJ\ "Zl i
) elm
a*.
Sin ce t h e diagram
U' (CPX)
= * • Κ (CP°°) ® Q
U' (CP°°)
, Κ (CP'l ® Q
is com m utative an d α (τ;) = η", we h a ve t h a t
f
p
*.y Ι Ρ! ί ff ir.r)) e Im a*
H
U
M
)
of so m e
a c t s t rivially on t h e fibration sp a c e of
Q [[ 1 η»]].
12
82
Ι· Μ. KRICEVER
From this and from ( 3) and ( 4) it follows that
where Ρ^ i s a polynomial in
Π ( Φ^) (Σ Ρ» · Ο η")*).
χ
m
>
η — 1 •' \ k
I
(η
ΐ)/ (χη + y).
τ
τ
Let 77 j be the c lo s e s t point to 1 at which there might be a pole of Φ
is , the c lo s e s t point to 1 of the form exp{2mr/ n),
(η); that
r < n and (r, n) = 1, for which
there i s a 7 . divisible by n. The function Φ χ (τ/ ) i s analytic in the dis c |τ;
|77j
1|; therefore the s e rie s Τχ
([X, S ] ) converges uniformly to it on every compact
subset of this d i s c From ( 5) it e as ily follows that the limit of Φ χ (.η) for η
is t s . H ence Φ (77) is analytic in the dis c \ η — l | < \ η2
1
'^
Τχ
1| <
η j ex
l|> and the s e rie s
([X, S ] ) converges uniformly on every compact subset of that dis c . Here η2
is
the next point at which there can be a pole of Φ χ {η). If we continue this proc ess we
obtain that Φ
( n) is analytic in the whole closed complex plane. Therefore it is a
constant. This concludes the proof of Lemma 2. 3.
Now we pa s s to the second part of the theorem. By Lemma 2. 2, Τχ
Φχ
( 1). S ince, by the previous part of the proof, Φ χ
( [ Χ] ) =
(η) i s constant, we have that
,,, (η) S TXiU ([M t]) Π
Remark. In what follows, all the limits are found under the assumption that χ +
y > 0. In the other c ase all the formulas are valid if we replace η » <» by i) >0.
Assume that / . > 0. Then
'ml
If we remember the definition of τχχ
((η), we obtain
x+ y
In an analogous way, for j
m
. < 0 we find that
+1
** *
V χ)
FORMAL GROUPS AND THE ATIYAH HIRZEBRUCH FORMULA
+ !f
(nii)
r
ml
/J \
ι
>*
1283
—(
Thus
+
—
£i
lim Φ,.ν (η) =»2 T x, y (l W,·]) * (_ yfJJ T x, y
where e^ i s the number of positive in tegers among the j
m i
and e~ i s the number of
negative on es, respectively.
Lei Σ . [ΛΓ ]I ] m ( C P ; . m ''*) be the part of β([Χ. S 1]) = I . [ M . ] n m ( C P / m ' ' ) e q u a l t 0
mi
J
*
*
k
mi
the bordism c la ss in R^ of the 5 bundle vg over a fixed submanifold Fg. Then for
; .
all the i. we have e.* = e + and f~ = f j . Since [ F ] = Σ . [Μ. ]Π [C P " " * ] , the proof
of Theorem 2.1 is complete.
§ 3 . Th e orientable case
We shall consider orientation preserving actions of compact L ie groups on manifolds
and vector bundles. All th e constructions and results of the precedin g section s for uni
tary action s automatically carry over to the present case; for th is reason we sh all re
strict ourselves to making statements only, with minimal explanations when n ecessary.
To each ch aracteristic c la ss y e Cl'(BSO) in the oriented cobordism of vector
bundles there corresponds a homomorphism of the Ω^ module of bordisms of oriented
C bundles over oriented C manifolds to the cobordism ring of the universal classify
ing space BG:
Theorem 3.1. For every characteristic class
lowing equality holds:
χ and every C bundle ξ, the fol
e (∆ο) %G ([ξ]) = 2 Ps: (e( vs)c) · X &c)).
s
The notation is the same as in Theorem 1.1, with the substitution of "orientable"
for "unitary" bundles (representations).
Let χ 0 be, a s before, th e "equivariant characteristic homomorphism" correspond
ing to the ch aracteristic c la ss 1 e £l°(BSO).
Let us consider an arbitrary orientable 5 manifold X. As we know, th e structure
group of the normal S bundle ν over a connected component F of the se t of fixed
poin ts under the action of S on X can be reduced to the unitary group and vs be
comes a complex S bundle ( see [10], §38). We choose the complex structure in vs
in such a way that the representation of S in the fibers h as th e form . {ηSl, / . > 0.
As before, we define a homomorphism of Ω^ modules
V»
1284
I. M. KRICEVER
where the summation is taken over all the collection s of nonnegative integers n. and
/ such that 2Σ . Qn • + I = n.
Theorem 3.2. There exists a homomorphism Ψ: R^ • Ω*[[κ]] 8>Q[u~ *] of Ω^ mod
ules such that f o j 3 ' coincides with the composite of the homomorphism \ Q
with the
homomorphism Ω*[[«]] » Ω*[[κ]]® Q[u~ ] . The values of Ψ on the generators of the
Ω^ module are given by the formula
Theorem 3.3. // α: G , » G is a homomorphism of Lie groups, then the diagram
is
commutative.
As in § 2, for each rational H irzebruch genus h: il^ • * Q we construct an equivar
iant H irzebruch genus hG: Ω^ • + K(BG) <8>Q.
The values of the classical Τ genus for y = 1 on almost complex manifolds
coincide with the signature of t h ese manifolds. Therefore, exactly as for Theorem 2.1,
one can prove
Theorem 3.4. For a connected compact Lie group G, the image of the homomor
C
phism Sign : Ω° » K(BG) ® Q belongs to the subring Q C K(BG) ® Q. For every or
iented S manifold X we have
Addendum. In a subsequent article, the proof of the following theorem will appear:
Theorem. // on a manifold X whose first Chern class Cj(X) € Η (Χ, Ζ) is div
isible by k there exists a nontrivial action of S , then / 4fe([x]) = 0.
The proof is based on "a n a lyt ic it y" arguments connected with the equivariant
series corresponding to the H irzebruch genus / 4fc, k = 2 3, . . . , given by the series
kt · el/ (ekt
1).
Received ll/ D E C / 73
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Translated by M. B. H ERRERO