An integral Riemann-Roch theorem for surface bundles

0.25 MB·

by  Ib Madsen

An integral Riemann-Ro
h theorem for surfa
e bundles
Ib Madsen
9 January 27, 2009
0
0
2
n 1 Introdu
tion
a
J This paper is a response to a
onje
ture by T. Akita about an integral Riemann-Ro
h theorem for

7 surfa
e bundles, [4℄, [5℄.
π: E → B g
2 Let be an oriented surfa
e bundle with
losed (cid:28)bers of genus , and equipped with
H(E) H1(E ;R)
b
] ag (cid:28)berwise metri
. Let be the asso
iated Hodge bundle with (cid:28)bers . It be
omes a
T -dimensional
omplexve
torbBundleBwUit(hg)the
omplexstru
tureindu
edfromtheHodgestaroperator,
A so is
lassi(cid:28)ed by a map from to . The element
n!ch ∈H2n BU(g);Z
. n
h
(cid:0) (cid:1)
t
a
m de(cid:28)nes a
hara
teristi

lass of the Hodge bundle and we set
s (E):=n!ch H(E) ∈H2n(B;Z).
[ n n (1.1)
(cid:0) (cid:1)
1 n n
They are torsion
lasses for even by [12℄, but not for odd where we shall
ompare them with the
v
standard Miller-Morita-Mumford
lasses
0
4 κn(E):=π∗ c1(TπE)n+1 ∈H2n(B;Z).
(1.2)
2 (cid:0) (cid:1)
4
TheGrothendie
kRiemann-Ro
htheorem,orthefamilyindextheoremofAtiyahandSinger,yields
.
1
0 the relation s2n−1(E)=(−1)n−1(Bn/2n)κ2n−1(E) H∗(B;Q),
in (1.3)
9
0
f. [28℄, [29℄. The Bernoulli numbers are de(cid:28)ned by the power series
: ∞
v z z
+ =1+ (−1)n−1B /(2n)!z2n.
Xi ez −1 2 nX=1 n
r
a Clearing denominators in (1.3), T. Akita made the following
onje
ture in [4℄:
H∗(B;Z)
Conje
ture (Akita). In ,
Denom(B/2n)s (E)=(−1)n−1Num(B /2n)κ (E).
2n−1 n 2n−1
n
The
onje
ture was veri(cid:28)ed in [5℄ for some spe
ial values of , but it turns out to be in
orre
t in
n = p
general. A
ounterexample is given in se
t. 7 below (for , an odd prime), but there are many
H∗(B;F )
p
other
ounterexamples and the
onje
ture is in
orre
t even in .
κ (E)
n
Onthepositivesidehowever,one
anrepla
ethestandard
lasses byanothersetofintegral
κ¯ (E) κ (E)
n n

hara
teristi

lasses whi
h I shall denote . In rational
ohomology they di(cid:27)er from only
by a sign: κ¯ (E)=(−1)nκ (E) H∗(B;Q)
n n
in , (1.4)
κ κ¯
n n
and moreover, Akita's
onje
ture be
omes
orre
t when we substitute the
lasses with . More
pre
isely we shall prove
1
H4n−2(B;Z)
Theorem A. (i) In ,
2Denom(B /2n)s (E)=2(−1)nNum(B /2n)κ¯ (E).
n 2n−1 n 2n−1
p κ (E)−(−1)nκ¯ (E)
n n
(ii) In
ohomology with -lo
al
oe(cid:30)
ients, the di(cid:27)eren
e is a torsion
lass of
p
order .
2
Remark 1. The extra fa
tor is unfortunate, and
ould possibly be removed with a more detailed

onsideration,
f. [17℄.
κ¯ (E) κ (E)
n n
The
lasses are not as simple to de(cid:28)ne as the standard
lasses . I owe the following
des
ription to Johannes Ebert, [14℄. Given an oriented surfa
e bu∂¯ndle (with
ompa
t (cid:28)ber), and
equipped with a (cid:28)berwise Riemannian metri
, one has the (cid:28)berwise -operator
∂¯: C∞(E;C)→C∞(E;T0,1E).
π
The target
onsists of se
tions in the
onjugate dual of the (cid:28)berwise tangent bundle, i.e.
TπE0,1 =HomC TπE,C .
(cid:0) (cid:1)
∂¯ C−H(E) ∂¯
The index bundle of is . More generally, one may twist with a
omplex ve
tor bundle
W E
on to get ∂¯ : C∞(E;W)→C∞(E;T0,1E⊗W).
W π
κ¯ (E)
n
The
lasses are then given by
κ¯ (E)=n!ch (index∂¯ ),
n n W
(1.5)
W = TπE −C index(∂¯ ) B K(B)
W
where and is the (analyti
) index bundle over . (Its
lass in is
W W
independent of
hoi
e oifnd
oenx(n∂¯e
t)ion on when is not holomorphi
.) The index theorem gives a
W
di(cid:27)erent des
ription of , better suited for our purpose.
g BDiff(F )≃BΓ Γ
g g g
Surfa
ebundleswith
onne
ted(cid:28)bersofgenus are
lassi(cid:28)edby ,where isthe
mapping
lass group. The proof of Theorem A depends on the homologi
al des
ription of the stable
Ω∞MT(2)= Ω∞CP∞
mapping
lass group in terms of the
obordism spa
e −1 from [25℄. In parti
ular
the Madsen-Weiss theorem is needed for the
ounterexample to Akita's
onje
ture.
π: E →B
An oriented surfa
e bundle indu
es a (cid:16)
lassifying map(cid:17)
α : B →Ω∞MT(2)
E
κ (E) κ¯ (E) s (E) κ κ¯ s
n n n n n n
andthe
lasses , and arepull-ba
ksofuniversal
lasses , and intheintegral
Ω∞MT(2)

ohomologyof . Theanalysisoftherelationshipbetweentheseuniversal
lassesusesin(cid:28)nite
loop spa
e theory and the theory of homology operations asso
iated with this theory.
The paper is divided up into the following se
tions:
1. Introdu
tion
2. Embedded surf¯a
e bundles
∂
3. The (cid:28)berwise -operator and its index
4. The main diagram
5. The key lemma
6. Proof of Theorem A
7. Dis
ussion of Akita's
onje
ture
p
8. Appendix on -power operations
2
2 Embedded surfa
e bundles
π: E →B
By Whitneys's embedding theorem, a smooth manifold bundle with
losed (cid:28)bers admits a
(cid:28)berwise embedding
i
E B×RN
π
pr
1
B
N N
for some , and the embedding is unique up to (cid:28)berwise isotopy if is su(cid:30)
iently large.
Thereareobviousadvantagesof
onsideringTeπmEbedded(cid:28)berbundles: pull-ba
ksbe
omefun
tional,
andthe(cid:28)berwise(orverti
al)tangentbundle inheritsaninnerprodu
tfromthestandardmetri
π: E →B
TonπEeu
lidian spa
e. Assuming further that J: TπEis→anToπrEiented surfa
e bundle, in+thπe sense that
is oriented, one gets a
omplex stru
ture by rotating ve
tors by 2.
The (cid:28)berwise normal bundle of the embedding,
NπE ={(z,v)∈E×RN |v ⊥TπE },
z
B×RN (z,v) i(z)+ π(z),v
maps to by sending to (cid:0) (cid:1). By the tubular neiDgh(NboπrEho)od theorem, we may
assume that this map restri
ts to an embedding of the unit disk bundle ,
f. [17, Ÿ3.2℄.
Given an oriented embedded surfa
e bundle, one gets a diagram of ve
tor bundles
NπE tˆ U⊥
2,N−2
p
(2.1)
t
E G(2,N −2)
,
G(2,N −2) 2 RN
where is the Grassmann manifold of oriented -planes in ,
U⊥ ={(V,v)|V ∈G(2,N −2), v ⊥V},
2,N−2
t(z)=TE ,
π(z)
tˆ(z,v)=(TE ,v).
π(z)
N =2n+2 n CPn G(2,2n)
Taking , we have the embedding of
omplex proje
tive -spa
e into by a
(2n−1)
-
onne
ted map, and a pull-ba
k diagram of ve
tor bundles
L⊥ U⊥
n 2,2n
CPn G(2,2n)
.
n t: E → G(2,2n) CPn
For large enough, is homotopi
to a map that fa
tors over , and (2.1)
an be
repla
ed by the diagram
NπE tˆ L⊥
n
p
(2.2)
E t CPn
.
NπE
One
onsequen
e of (2.2) is that we may assumEe t⊂haBt ×Rn+is2a
omplex ve
torbundle. Alternatively
a
omplex struN
tuπrEe
an bTeπsEeen⊂aBs×foTlloRwns+:2if is a (cid:28)berwqi∗s(eNeπmEb⊗edRdCin)g with vqerti
al
normalbundle ,then has
omplexnormalbundle ,where isthe
3
TπE E ⊂TπE ⊂B×TRn+2 TπE⊕(NπE)⊗RC
bundleproje
tionof , [7℄,[9℄,and hasnormalbundle
; this is a
omplex ve
tor bundle.
π: E →B
E ⊂FoBr×theRr2ens+t2ofthepaper,asurfa
ebundle willmeananembeddedorientedsurfa
ebundle
NπE →B×R2nw+i2th
ompa
t (cid:28)ber and
omplex verti
al normal bundlDe,(sNu
πhEt)hat the standanrd map
reEstr⊂i
tBs t×oRan2ne+m2bedding of the uniEt d⊂is
Bb×unRd2lne+s4 . The integer is not
part of the stru
ture: is identi(cid:28)ed with .
The di(cid:27)eomorphism
lasses of su
h surfa
e bundles are in bije
tive
orresponden
e with the set
[B,BDiff(F)] F Diff(F)
of homotopy
lasses , where is the (cid:28)ber surfa
e and denotes the topologi
al
group of orientation preserving di(cid:27)eomorphisms,
f. Ÿ2.1 of [18℄.
For su
h a surfa
e bundle, we
an apply the Pontryagin-Thom
onstru
tion on the embedding
D(NπE)֒→B×R2n+2,
giving the map c : B ∧S2n+2 →Th(NπE):=D(NπE)/∂ .
E +
(2.3)
π K
∗
There is an indu
ed push-forward map in -theory de(cid:28)ned by the diagram
K(E) Φ K Th(NπE)
(cid:0) (cid:1)
π∗ e (cE)∗
K(B) Φ K(B ∧S2n+2)
∼= + ,
Φ K e
where denotes the -theory Thom isomorphism. It depends on the
hoi
e of Thom
lass; we use
the
lascshfro:mK[(6B,)Ÿ2→.7℄H. 2n(B;Q) n
n
Let be the 'th
omponent of the Chern
hara
ter,and
s =n!ch : K(B)→H2n(B;Z)
n n
the a

ompanying map into integral
ohomology. The
lasses that appear in Theorem A have the
following alternative des
ription
s (E)=(−1)ns π (1) ,
n n ∗
κ¯ (E)=s π ([T(cid:0)πE]−(cid:1)1) , (2.4)
n n ∗
(cid:0) (cid:1)
TπE (TπE,J) HomC(TπE,C)
where is
onjugateto andisomorphi
totheverti
al
omplex
otangentbundle .
The relationship between (2.4) and the des
ription given in the introdu
tion is the index theorem
[9℄. This is dis
ussed in the next se
tion.
¯
∂
3 The (cid:28)berwise -operator and its index
∂¯
This short se
tion explains the family index theorem for the (cid:28)berwise -operator. This is well-known
for the experts, but I will in
lude some details for the more inexperien
ed reader.
V J: V →V J2 =−id
Let beave
torspa
ewithinnerprodu
tandisometri

omplexstru
ture , .
φ7→iφJ−1 HomC(V,C) ±1
The involution on de
omposes into eigenspa
es
HomR(V,C)=HomC(V,C)⊕HomC(V,C)
V =(V,J) V =(V,J−1) HomR(V,R)
where on the right hand side and . The subspa
e sits diagonally
in the dire
t sum, and the proje
tions de(cid:28)ne isomorphisms
HomR(V,R)∼=HomC(V,C), HomR(V,R)∼=HomC(V,C).
(3.1)
4
φ 7→1/2(φ+iφJ−1) φ 7→1/2(φ−iφJ−1)
The two isomorphisms are given by and , respe
tively. The
h, i V V HomR(V,R)
inner produ
t on identi(cid:28)es with and the
ombined isomorphism
V −−∼=→HomR(V,R)−−∼=→HomC(V,C),
(3.2)
w ∈V 1/2(φ −iφ J−1) φ (v)=hv,wi
w w w
whi
h sends to with TπE , is a
omplex isomorphism.
Applied (cid:28)berwise to the verti
al tangent bundle of an embedded surfa
e bundle, de(cid:28)ne
Tπ1,0E :=HomC(TπE,C), Tπ0,1E :=HomC(TπE,C)
so that C∞(E,TπE)=C∞(E,T1,0E)⊕C∞(E,T0,1E).
π π
C∞(E,C) C∞(E,TπE)
The (cid:28)berwise exterior di(cid:27)erential from into de
omposes a

ordingly. The
se
ond
omponent is ∂¯: C∞(E,C)→C∞(E,T0,1E),
π
F π: E →B ∂¯φ= ∂φdz¯
whi
h on a (cid:28)ber of is given lo
ally by ∂z¯ with respe
t to a
omplex
oordinate
F J
on
ompatible with .
W E
More generally, if is a
omplex ve
tor bundle on then (possibly after
hoi
e of
onne
tion)
one has the asso
iated operator
∂¯ : C∞(E,W)→C∞(E,T0,1E⊗W).
W π
σ(∂¯ )
W
The symbol is identi(cid:28)ed as
σ(∂¯W)=Φ([W])=λTπE ·[W]∈K˜ Th(TπE) ,
(3.3)
(cid:0) (cid:1)

f. [8, Ÿ4℄. This
lass is independent of
hoi
e of
onne
tion. The (virtual) index bundle is
index(∂¯ )=ker(∂¯ )−cok(∂¯ ).
W W W
K(B)
It de(cid:28)nes a
lass in , and the index theorem of [9℄ asserts that
[index(∂¯ )]=π σ(∂¯ ) ∈K(B).
W ! W
(3.4)
(cid:0) (cid:1)
π : K˜ Th(TπE) →K(B) π : K(E)→K(B)
! ∗
Thepush-forward of[9℄isrelatedtoour viatheformula
(cid:0) (cid:1)
π ([W])=π Φ([W]) .
∗ !
(3.5)
(cid:0) (cid:1)
Hen
e (3.4) has the alternative formulation:
[index∂¯ ]=π ([W]),
W ∗
(3.6)
and the
lasses of (2.4) are given by
s (π [W])=s (index∂¯ ),
n ∗ n W
(3.7)
[W]=1 [W]=[TπE]−1 index(∂¯)
for and ,respe
tively. Wehavelefttoidentify intermsoftheHodge
bundle. The result is:
ker∂¯=B×C
Lemma 3.8. (i) ,
cok∂¯=H(E) π (1)=1−[H(E)]
∗
(ii) , the Hodge bundle, and .
5
F π: E → B ∂¯ F C
|F
Proof. On a (cid:28)ber of , the kernel of
onsists of the holomorphi
maps from to
F
wh∂¯i
h, sin
e is a
losed Riemannian surfa
e, are the
onstant maps. This proves (i). The
okernel
|F
of is
cok∂¯ =H1(F;O),
|F
O
where is the sheaf of holomorphi
maps, see e.g. [15, Thm. 15.14℄. On the other hand, by Serre
duality H1(F;O)=HomC Ω(F),C
(cid:0) (cid:1)
Ω(F) 1
where is the ve
tor spa
e of holomorphi
-forms, [15, Thm. 17.9℄.
∗ H1(F;R) 1
WeusedtheHodge -operatortogivethespa
e ,ofharmoni
-forms,a
omplexstru
ture,
isometri
w.r.t. the inner produ
t given by Poin
aré duality. The isomorphism
H1(F;R)−−∼=→Ω(F),
1 φdx+ψdy (φ−iψ)dz
givenlo
allyby sendingthe harmoni
-form to , is
onjugate linear. Thisyields
the
omplex linear isomorphism
H1(F;R),∗ ∼=HomC Ω(F),C .
(cid:0) (cid:1) (cid:0) (cid:1)
Sin
e the Hodge bundle is Princ(E)× H1(F;R),∗
Diff(F)
(cid:0) (cid:1)
(cid:3)
this proves(ii).
4 The main diagram
E ⊂B×R2n+1
Let be an embedded, oriented surfa
e bundle as in Ÿ1,
lassi(cid:28)ed by
NπE tˆ L⊥
n
E t CPn
,
and indu
ing the map tˆ: Th(NπE)→Th(L⊥).
n
Let ω: Th(L⊥)→Th(L ⊕L⊥)=CPn ∧S2n+2
n n n + (4.1)
L π (1) π (TπE −1) K(B)
n ∗ ∗
be the in
lusion along the zero se
tion of . The elements and of are
expli
itly given by the formulas
π∗(1)=Φ−B1c∗Etˆ∗(λL⊥n),
π∗(TπE−1)=Φ−B1c∗Etˆ∗ω∗ΦCPn(L¯n), (4.2)
Φ : K(X) → K˜(X ∧ S2n+2) X
X +
wch:erBe ∧S2n+2 →Th(NπE) is the Thom isomorphism for the trivial bundle on , and
E +
is the
ollapse map of (2.3). For the se
ond formula in (4.2), note that
ω∗(λL+L⊥)=(1−L)λ⊥L. λL⊥ ω∗ΦCPn(L¯n)
Wenowdes
ribethekeyrelationbetweentheelements nand whi
heventuallyleads
to the proof of Theorem A.
6
n W X
Given an -dimensional
omplex ve
tor bundle on ,
̺k(W)∈K(X)⊗Z[1]
k
K
is the -theory
hara
teristi

lass de(cid:28)ned by
ψk(λ )=kn̺k(W)λ ,
W W
ψk k
where is the 'th Adams operation. Then
̺k(V ⊕W)=̺k(V)̺k(W),
̺k(L)= 1(1+L+···+Lk−1), L
k a line bundle.
rk
De(cid:28)ne the operation by the diagram
K(X) Φ K˜(X ∧S2)
∼= +
rk ̺k
(4.3)
K(X)⊗Z[1] 1+Φ 1+K˜(X ∧S2)⊗Z[1]
k ∼= + k .
It is additive, rk(V ⊕W)=rk(V)+rk(W),
L L¯
and for a line bundle it is given by the formula (with the
onjugate line bundle)
Lk−2+2Lk−3+···+(k−2)L+(k−1)
rk(L¯)= .
1+L+···+Lk−1 (4.4)
Φ λC = L1−1 K˜(S2)
This follows easily upon using that in (4.2) is (exterior) multipli
ation by in ,
together with the equation
(L−1) Lk−2+2Lk−3+···+(k−1) =(1+L+···+Lk−1)−k,
(cid:0) (cid:1)

f. [23, Prop. 6.2℄. Sin
e ω∗(λLn⊕L⊥n)=(1−Ln)λL⊥n, we get
(k−nψk−id)(λL⊥n)=ω∗ rk(L¯n)λLn⊕L⊥n . (4.5)
(cid:0) (cid:1)
(Z×BU)[1]
This relation
an also be eKxp(rXes)s⊗edZa[s1t]he homotopy
ommutative diagrambelow, where k
is the
lassifying spa
e of k , i.e.
K(X)⊗Z[1]= X,(Z×BU)[1]
k k
(cid:2) (cid:3)
X
for
ompa
t . The diagram is
Th(L⊥) ω CPn ∧S2n+2 L¯n∧S2n+2 (Z×BU)∧S2n+2
n +
λL⊥n rk∧S2n+2 (4.6)
Z×BU k−nψk−id (Z×BU)[1] ε (Z×BU)[1]∧S2n+2
k k ,
ε: (Z×BU)∧S2n+2 →BU λCn+1 ∈K˜(S2n+2)
with indu
ed from multipli
ation with .
In order to make the above diagram independent of the embedding dimension, we pass to spe
tra
and their asso
iated in(cid:28)nite loop spa
es,
f. [3℄, [2℄.
7
A={A ,ε }
n n
We remember that for a (pre) spe
trum the asso
iated in(cid:28)nite loop spa
e is
Ω∞A=colimΩnA
n
ΩnA → Ωn+1A ε′ : A → ΩA A Ω
where the maps n n+1
omes from the adjoint n n n+1. If is an ( -)
ε′ : A →ΩA Ω∞A ≃ A
spe
trum, i.e. if n n n+1 is a homotopy equivalen
e, then 0. The spe
tra relevant
Σ∞Y KU MT(2) (2n+2)
to us are , and . They have 'nd spa
es:
(Σ∞Y) =Y ∧S2n+2,
2n+2
KU =Z×BU,
2n+2
MT(2) =Th(L⊥).
2n+2 n
Ω2n+2(−) n
Applying to (4.6) and taking
olimit over leads to the following homotopy
ommutative
Q(Y) = Ω∞Σ∞Y X =
+
diagram of in(cid:28)nite loop spa
es, where we use the standard notations: and
X⊔{+}
,
Ω∞MT(2) ω Q(CP∞) Q(L¯) Q(Z×BU)
+
λ−L Q(rk) (4.7)
Z×BU kψk−id Z×BU[1] E Q (Z×BU)[1]
k k .
(cid:0) (cid:1)
E: Q(Z×BU)→Z×BU k
Themap Ω2n(Z×BU)≃Z×BU, andits lo
alisedversionwΩith inverteAd,=is{aA
o,nεse}quen
eofBott
n n
periodi
ity . More generally, for any -spe
trum , the maps
Ωn(A ∧Sn)→ΩnA ←−≃−A
0 n 0
E: Q(A )→A
0 0
determines a (weak) homotopy
lass .
The
omposition tˆ◦c : B ∧S2n+2 →Th(NπE)→Th(L⊥)
E + n
(2n+2)
of an embedded surfa
e bundle is the 'nd
omponent of a map of spe
tra
c : Σ∞(B )→MT(2).
E +
Its adjoint is α : B −i−B→Q(B )−−Ω−∞−c−E→Ω∞MT(2),
E +
(4.8)
i
B
where is the in
lusion.
With these notations, the two elements of (4.2) are represented by the homotopy
lasses
B −α−E→Ω∞MT(2)−λ−−−L→Z×BU,
B −α−E→Ω∞MT(2)−ω→Q(CP∞)−−Q−(L−¯→) Q(Z×BU)−−E→Z×BU. (4.9)
+
Theorem A is proved by evaluating (4.7) on
ohomology, but to a

omplish this one needs to
examine the diagram
Q(Z×BU) E Z×BU
Q(rk) rk
(4.10)
Q (Z×BU)[1] E (Z×BU)[1]
k k
(cid:0) (cid:1)
rk
on the
ohomologi
allevel. Unfortunately, (4.10) is not a homotopy
ommutative diagram: is only
twi
e deloopable but not an in(cid:28)nite loop map.
BU ={0}×BU ⊂Z×BU
We may restri
t (4.10) to the zero
omponent , and lo
alise at a prime
p (k,p)=1
with .
8
(k,p)=1
Key Lemma 4.11. For , the
ohomologi
al diagram
PrimH∗ Q(BU );Z E∗ PrimH∗(BU ;Z)
(p) (p)
(cid:0) (cid:1)
Q(rk)∗ (rk)∗
PrimH∗ Q(BU );Z E∗ PrimH∗(BU ;Z)
(p) (p)
(cid:0) (cid:1)
is homotopy
ommutative.
Prim( )
Here denotes the primitive elements in the Hopf algebras, e.g.
PrimH2n(BU ;Z)=Z hs i, s =n!ch ,
(p) (p) n n n
Z ⊂ Q p
(p)
where are the fra
tions with denominator prime to . The lemma is proved in the next
se
tion.
5 The key lemma
This se
tion deals with the non-
ommutativity of diagramp (4.10). Our meth̺okd is torkuse t(hke,pA)r=tin1-
Hasselogarithmof[33℄. Tothis endwesingleoutaprime andonly
onsider and for .
K (X)=K(X)⊗Z
(p) (p)
Let us write . We have the group homomorphism
̺k: K˜ (X)→1+K˜ (X)
(p) (p)
1
where the group stru
ture on tKh˜e(Xta)rgetBiUs⊕ten=so{r0}pr×odBuU
t of virtual ve
tor bundles of dimension .
The represeKntingspa
efor bu⊕is . Thisisthein1(cid:28)+niKt˜e(lXoo)p sBpaU
e⊗a=sso{1
i}a×teBdUto
the
onne
ted -theoryspe
trum of[3℄. Therepresentingspa
efor is .
bu⊗
Thisisthe in(cid:28)nite loop spa
eofaspe
trum ,
onstru
tedfrom theabstra
ttheoryof in(cid:28)nite loop
spa
es [11℄, [26℄, [31℄. The map ̺k: BY⊕ →BU⊗
(p) (p)
bu⊕ →bu⊗
is an in(cid:28)nite loop map by [24℄, i.e. it lifts to a map of spe
tra (p) (p).
The Artin-Hasse logarithm L : 1+K˜ (X)→K˜ (X)
(p) (p) (p)
X
is for
ompa
t de(cid:28)ned by the formula
∞
1
L (1−x)=− θpt(xn),
(p) n
(nX,p)=1 Xt=0
θpt: K˜ (X)→K˜ (X)
(p) (p)
where is the operation
θpt(x)= 1 xpt −ψp(xpt−1) , t>0
pt
(cid:0) (cid:1)
θ1(x) = x K˜ (X) ψp ψp(x) ≡ xp (mod p)
(p)
and . It exists in be
ause is multipli
ative and , and it is
K˜(BU)
uniquely de(cid:28)ned sin
e is torsion free. Note that rationally,
ψp
L (1−x)= −1 log(1−x)∈K˜(X)⊗Q.
(p) (cid:18) p (cid:19)
9
L l
(p) (p)
It is the double loop of we are interested in, i.e. in the map de(cid:28)ned by the diagram
1+K˜ (X ∧S2) L(p) K˜ (X ∧S2)
(p) (p)
∼= 1+Φ ∼= Φ
K˜ (X) l(p) K˜ (X)
(p) (p)
.
l
(p)
Lemma 5.1. The map is given by the formula
l (x)=x+ψp(x).
(p)
K
Proof. All produ
ts vanish in redu
ed -theory of a suspension, so
∞
−L(p)(1−xλC)= ψpt(xλC)=xλC+ p1ψp(xλC)
Xt=0 =xλC+ψp(x)̺p(1)λC
= x+ψp(x) λC
(cid:0) (cid:1)
K (X)
(p)
Stri
tlXy s=peBakUing, this
al
ulation only makes sense when is torsion free. But this is the
as(cid:3)e
when . In general the formula follows by naturality from this
ase.
Based on the
riteria from [24℄, tom Die
k showed in [33℄ that
L : BSU⊗ →BSU⊕
(p) (p) (p)
L ◦̺k BSU⊕
isanin(cid:28)niteloopmap. The
omposite (p) from (p) toitselfisthenalsoin(cid:28)nitelydeloopable.
Taking double loops implies that
BU⊕ −−rk→BU −−l(−p→) BU⊕
(p) (p) (p)
is an in(cid:28)nite loop map, and the diagram
Q(BU ) Q(rk) Q(BU ) Q(l(p)) Q(BU )
(p) (p) (p)
E E (5.2)
BU rk BU l(p) BU
(p) (p) (p)
l
(p)
is
onsequentlyhomotopy
ommutative. Lemma5.1showsthat indu
esisomorphismonhomotopy
groups, so is a homotopy equivalen
e.
l Q(l ) F
(p) (p) p
Lemma 5.3. The maps and indu
e the identity on
ohomology with
oe(cid:30)
ients.
Proof. It su(cid:30)
es to
he
k that
l : H (BU ;F )→H (BU ;F )
(p)∗ ∗ (p) p ∗ (p) p
H Q(X);F H (X,F )
∗ p ∗ p
is the identity, sin
e (cid:0) ψp (cid:1) is a fun
tor of pn .H2n(BU(p);Z)
The Adams operation indu
es multipli
ation by on , so indu
es the zero map
H˜ (BU ;F ) l =id (cid:3)
∗ (p) p (p)
on . By Lemma 5.1, .
10