Table Of ContentCANNIBALISTIC CLASSES OF STRING BUNDLES
GERDLAURESANDMARTINOLBERMANN
7
1
0 Abstract. We introduce cannibalistic classes for string bundles with values
2 in TMF with level structures. This allows us to compute the Morava E-
homology of all maps from the bordism spectrum MString to TMF with
n
levelstructures.
a
J
5
] 1. Introduction and statement of results
T
Suppose E is a cohomology theory and V is a bundle over a space X which is
A
equipped with a Thom class τ with respect to E. Then for each stable operation
.
h ψg in E-theory one obtains a cannibalistic class θg(V) by the formula
t
a (1.1) ψg(τ) = θg(V)τ.
m
For example, for singular cohomologywith coefficients in F the cannibalistic class
2
[
θk associatedto the SteenrodsquareSqk is givenby the kth Stiefel-Whitney class.
1 Another example arises in real and complex K-theory for Adams operations (see
v [Bot62]). In the complex case, the cannibalistic classes of a line bundle L can be
7
computed via the formula
8
2 ψg(x)
(1.2) θg(L) =
1 x
0
where x is the Euler class. The cannibalistic classes play an important role in the
.
1 investigation of bordism invariants and spherical fibrations.
0
In this paper, we are interested in cannibalistic classes for string bundles with
7
valuesinthespectrumoftopologicalmodularformsorinMoravaE-theory. Wewill
1
: showhowtheyarerelatedtothestringcharacteristicclassesdefinedin[LO16]. This
v
enables us to compute the E-homology of maps form the string bordism spectrum
i
X to the spectrum of topological modular forms with level structures.
r We will now describe the results in more detail. In the K(2)-local category at
a
the prime 2 the spectrum TMF(3) of topological modular forms with respect to
Γ(3) coincides with the Morava E-theory spectrum E and there is a continuous
2
action of the Morava stabilizer group G on E . Moreover, string manifolds have
2
a natural Thom class with values in TMF(3) and in all of its fixed point spectra
for finite subgroups of G. This is known as the Witten orientation [AHR]. Hence,
therearestablecannibalisticclassesθg forstringbundlesandforallelementsofG.
In [LO16], all TMF (3)-characteristic classes for string bundles have been com-
i
puted for i=0,1. It turned out that they are generatedby Pontryaginclasses and
one more class r which, in the complex setting, measures the difference between
the complex orientation from the elliptic curve and the Witten orientation. The
Date:January6,2017.
2000 Mathematics Subject Classification. Primary55N34;Secondary55P20, 22E66.
1
2 GERDLAURESANDMARTINOLBERMANN
following formula relates this class for U 5 -bundles and the cannibalistic class θC
h i
of the complex orientation:
(1.3) θg = qgrdet(g)θg.
0 C
Here, qg is a specific SU-characteristic class which describes the difference be-
0
tween the cubical structures of the underlying elliptic curves and its perturbation
by g. It will be computed in more detail in section 3. The complex cannibalistic
classθg satisfiestheformula1.2above. Wewillproveasplittingprincipleforstring
C
bundles which reduces the calculation of the string cannibalistic classes to (1.3).
The formula allows us to compute the Morava E-homology of all maps from
the Thom spectrum MString to TMF (3). In [LO16] and [Lau16] it was shown
i
that the cohomology of MString is topologically freely generated by TMF (3)-
i
Pontryaginclasses and the class r described above. The precise formulas in terms
of the dual generators are given in section 4.
TheseresultswillproveusefulintheinvestigationofindecomposablesofString-
bordism. We think that in the K(2)-localcategory there is an additive splitting of
the form
MString =TMF ΣniTMF
∼ ∨ Γi
i_I
∈
for some level structures Γ . The computations in this work are all very explicit
i
andcanhelptofindthecorrectindexsetI. However,theonlyclasswhichprevents
us to set up the splitting formula is the class q described above. It seems to be
0
hard to write down in a closed form in high dimensions.
Acknowledgements. TheauthorswouldliketothankCraigWesterlandandThomas
Nikolaus for helpful discussions.
2. TMF with level structure and characteristic classes
Inthissectionwerecallthedefintionofthecharacteristicclassesinthespectrum
of topological modular forms with level structures. We refer the reader to the
articles [MR09], [Lau16] and [LO16] for an account on the spectra
T =TMF (3)
i i
for i ,0,1 .
∈{∅ }
Consider the supersingular elliptic curve in Weierstrass from
C : y2+y =x3
over F . There is a Lubin-Tate spectrum E (also called Morava E-theory) associ-
4 2
ated to the formal group Cˆ with coefficients
E =W(F )[[u ]].
2 4 1
∗
Let G be the automorphism group of Cˆ Spec F and let G be the one of
2 4
→
C Spec F (i.e. automorphisms are commutative squares). There is an isomor-
4
→
phismG =S ⋊Gal(F :F ), where the subgroupS correspondsto commutative
2 ∼ 2 4 2 2
diagrams
Cˆ g //Cˆ
(cid:15)(cid:15) (cid:15)(cid:15)
Spec F = //Spec F ,
4 4
CANNIBALISTIC CLASSES OF STRING BUNDLES 3
and the Galois group acts via the Frobenius, which gives a commutative diagram
Cˆ σ // Cˆ
(cid:15)(cid:15) (cid:15)(cid:15)
Spec F σ //Spec F .
4 4
Abstractly, the group S can be described as the group of units in the maximal
2
order
End(Cˆ)=Ø =Z ω,i,j,k
2 2
{ }
of the 2-adic quaternion algebra
1
End(Cˆ) =D =Q 1,i,j,k .
2 2
(cid:20)2(cid:21) { }
Here ω = 1( 1 i j k) such that ω2 = ω 1,ω3 = 1. One computes
2 − − − − − −
that ωiω 1 = j,ωjω 1 = k,ωkω 1 = i. Note that we do not make a notational
− − −
difference between the endomorphisms of Cˆ and the endomorphisms of the Honda
formalgrouplaw; theseringsareisomorphic(see also[Bea15]). Adetailedaccount
on the Morava stabilizer group can be found in the appendix.
ThegroupGisamaximalfinitesubgroupofG andhasorder48. Theautomor-
2
phisms 1,ω,i,j,k arise fromautomorphismsof C overthe identity ofF =F [α]:
4 2
−
1 corresponds to mapping each point on the curve to its negative, i.e.
• −
x x,y y+1,
7→ 7→
ω corresponds to x αx,y y, and to the automorphism g(x) = αx of
• 7→ 7→
the formal group,
i corresponds to x x+1,y x+y+α.
• 7→ 7→
The Frobenius map σ is the generator of Gal(F : F ), it acts via x x2,y y2
4 2
7→ 7→
on C as an automorphism over the Frobenius map of F .
4
Note that G=G ⋊Gal(F : F ), where G is generated by 1,ω,i,j,k, and
∼ 24 4 2 24 −
G =Q ⋊C , where C is generated by ω and Q = 1, i, j, k .
24 ∼ 8 3 3 8 {± ± ± ± }
The group of F -points of C is isomorphic to (Z/3)2. Via the induced action of
4
G we obtain an isomorphism
G=GL (Z/3).
∼ 2
Choosing as generators the F -points (0,0) and (1,α), this isomorphism sends
4
1 0 1 1 0 1 1 0
1 − , ω , i − , σ .
− 7→(cid:18) 0 1(cid:19) 7→(cid:18)0 1(cid:19) 7→(cid:18)1 0 (cid:19) 7→(cid:18)0 1(cid:19)
− −
InparticularSL (Z/3)correspondstothegroupG ,thesubgroupG ofGL (Z/3)
2 24 0 2
which fixes the subgroup Z/3 0 corresponds to the group generated by 1,ω,σ
× −
and the group G which fixes the point (1,0) of (Z/3)2 corresponds to the group
1
generated by ω and σ.
Theorem 2.1. [Beh07, Sect.5] [Ban14, Sect. 3.1]. Let K(2) be the second Morava
K-theory at the prime 2. There is a continuous action ψ of G on E and canonical
2
4 GERDLAURESANDMARTINOLBERMANN
isomorphisms
L TMF = EhG,
K(2) ∼ 2
L T = EhG0,
K(2) 0 ∼ 2
L T = EhG1,
K(2) 1 ∼ 2
L T = E .
K(2) ∼ 2
In the sequel we will work in the K(2)-local category and we will omit the
localizationfunctorfromthenotation. Thefollowingresultcanbefoundin[DH95]
(see also [BOSS]).
Proposition2.2. Anelementf π (E T )gives risetothecontinuousfunction
k 2 i
∈ ∧
from G /G to π E which sends g to the composite of f with
2 i k 2
1 ψg
E T ∧ // E T // E E //E .
2 i 2 i 2 2 2
∧ ∧ ∧
This map is an isomorphism
φ:π (E T ) ∼= Map (G /G ,π E )
∗ 2∧ i −→ cts 2 i ∗ 2
In this notation, the operations satisfy
(1 ψν)f(g) = f(gν)
∧
(ψν 1)f(g) = ψνf(ν 1g)
−
∧
Locallyatthe prime2,forthetheoryT ithasbeenshownin[Lau16]thatthere
1
are unique classes p T4iBSpin with the following property: the formal series
i ∈ 1
p =1+p t+p t2... is given by
t 1 2
m
(1+tρ (x x ))
∗ i i
Yi=1
when restricted to the classifying space of each maximal torus of Spin(2m). Here,
ρ is the map to the maximal torus of SO(2m) and the x (and x ) are the first
i i
T -Chern classes of the line bundles L (resp. L ). The classes p freely generate
1 i i i
the T cohomology of BSpin, that is,
1
T1∗BSpin∼=T1∗[[p1,p2,...]].
It was shown in [Lau16] that the Kitchloo-Laures-Wilson sequence implies an iso-
morphism of algebras
T1∗BString ∼=T1∗[[r,p1,p2,...]]
where p ,p ,... are the Pontryaginclasses coming fromBSpin and r restricts to a
1 2
topological generator of degree 6 in the K(2)-cohomology of K(Z,3).
In[LO16]theresulthasbeenusedtostudytheT cohomologyofBString using
0
equivariantmethods. ThetheoryT admitsthestructureofaRealspectruminthe
1
senseofAtiyah,Kriz,Hu etal. (compare[Ati66][HK01]). This meansthat there is
a Z/2-equivariant spectrum (“the Real theory”) whose non-equivariant restriction
(“the complex theory”) is T and whose fixed point spectrum (“the real theory”)
1
is T0. They were used in [LO16] to show that there are classes πi ∈ T0−32iBSpin
which lift the products v6ip for the T -Pontryagin classes p . Moreover, we have
2 i 1 i
an isomorphism
T0∗BSpin∼=T0∗[[π1,π2,...]]
CANNIBALISTIC CLASSES OF STRING BUNDLES 5
Inordertoobtainaclassr whichisalreadydefinedintherealtheoryT onehasto
0
provide a more geometric construction. Here, the theory of cubical structures on
elliptic curves comes in and furnishes a construction of the Witten orientation in
[AHS01]. It turns out that a convenient choice of a generator is possible in BU 6
h i
where we define
re σ
∗
r := T BU 6
U x ∈ 1∗ h i
as the difference class which compares the Witten orientation
re σ
MU 6 MString T
1
h i→ →
withthecomplexorientationx T MU 6 . Foreachstablecomplexvectorbundle
∈ 1∗ h i
over a finite CW-complex X equipped with a lift of its structure map X BU to
→
ξ :X BU 6 , we obtain a characteristic class r (ξ) T X by pulling back r .
→ h i U ∈ 1∗ U
The spaceBU 6 hasanH-spacestructure induced fromBU, andthe inclusion
h i
of the homotopy fiber of BU 6 BSU is an H-space map j :K(Z,3) BU 6 .
h i→ → h i
For two bundles ξ : X BU 6 , η : X BU 6 , we obtain a direct sum ξ η :
→ h i → h i ⊕
X BU 6 such that
→ h i
r (ξ η)=r (ξ)r (η),
U U U
⊕
since both Thom classes are multiplicative.
Remark 2.3. When restricted to K(Z,3) the class r := r (j) T K(Z,3)
K U ∈ 1∗
can be described in a second way, which was suggested to the authors by Thomas
Nikolaus: in the diagram
j
(2.1) K(Z,3) // BU 6 //BSU
h i
= re re
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
K(Z,3) i //BString // BSpin,
pull the universal bundle over BSpin back and take Thom spaces everywhere to
obtain
Tj
(2.2) K(Z,3) // MU 6 // MSU
+
h i
= re re
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
K(Z,3) Ti //MString //MSpin,
+
Herethe compositionofthe complexorientationwithTj factorsthroughMSU,so
K(Z,3) MU 6 x T is the unit element in T K(Z,3). Thus the composition
+ → h i→ 1 1∗
K(Z,3) MU 6 re MString σ tmf T T
+ 0 1
→ h i→ → → →
is equal to the pull-back r =j re∗σ: K(Z,3) BU 6 T . In particular we
K ∗ x + → h i→ 1
seeanaturalliftofr toaclassinT K(Z,3),andinfactevenalifttotmf K(Z,3).
K 0∗ ∗
Note that
(2.3) (Ti) σ = (Tj) (re σ)=r .
∗ ∗ ∗ K
The complex conjugation on BU induces an involution on BU 6 . We denote
h i
the composition of ξ with this complex conjugation on BU 6 by ξ and denote the
h i
6 GERDLAURESANDMARTINOLBERMANN
standard involution on T by ψ 1. In [LO16] it is shown that r can be turned
1 − U
into a Z/2-equivariant map. It follows that
r (ξ)=ψ 1(r (ξ)).
U − U
Since c:BString BU 6 maps to the fixed points of complex conjugation on
→ h i
BU 6 ,the pull-backr =r (c)admitsalifttothefixedpointspectrumT ofψ 1.
U 0 −
h i
By pulling back r along the classifying map, there is a natural stable class
r(ξ) T0X
∈ 0
for every string bundle ξ over X.
Theorem 2.4. [LO16] The class r has the following properties:
(i) r is multiplicative: r(ξ η)=r(ξ) r(η).
⊕ ⊗
(ii) There is an isomorphism
T0∗[[r˜,π1,π2,...]] T0∗BString
−→
where r˜= r 1 is the reduced version of the class r corresponding to the
−
universal bundle over BString.
(iii) In terms of the Chern character of its elliptic character λ (c.[Mil89]) at
the cusp it is given by the formula
∞
Φ(τ,x ω)
i
ch(λ(r (ξ)))= −
U
Φ(τ, ω)
Yi −
where the x are the formal Chern roots of ξ C, ω =2πi/3 and Φ is the
i
⊗
theta function
Φ(τ,x) = (ex/2 e−x/2) ∞ (1−qnex)(1−qne−x)
− (1 qn)2
nY=1 −
∞ 2
= x exp( G (τ)x2k).
2k
− (2k)!
Xk=1
Remark 2.5. Since the diagram
(1,conj)
(2.4) BU 6 // BU 6 BU 6
h i h i× h i
re
⊕
(cid:15)(cid:15) (cid:15)(cid:15)
c
BString // BU 6
h i
commutes, we obtain the formulas
(2.5) r (c re) = r(re)=r ψ 1r ,
U U − U
◦ ·
(2.6) r(i) = r(re j)=r (j) ψ 1r =r2 ,
◦ U · − K K
where the last equality holds since r comes from a T -cohomology class and is
K 0
thus Z/2-invariant.
Theorem 2.6. (i) Let b π (E BS1) be the dual to the power ci of the
i ∈ 2i 2∧ 1
first Chern class of the canonical line bundle. Denote its image under the
map induced by the inclusion of the maximal torus
ι:BS1 //BSpin(3) //BSpin
CANNIBALISTIC CLASSES OF STRING BUNDLES 7
by the same name. Then
π E BSpin =π E [b ,b ,b ,...].
∗ 2∧ + ∼ ∗ 2 8 16 32
(ii) Let d π (E K(Z,3)) be a class which restricts to a generator in d
0 2
∈ ∧ ∈
π (K(2) K(Z,3)) as divided power algebra. Then π (E K(Z,3)) is
0 2
∧ ∗ ∧
generated by d.
(iii) Denote a lift of b to π (E BString) by a . Then we have
i 2i 2 i
∧
π E BString =π (E K(Z,3))[a ,a ,a ,...].
∗ 2∧ + ∼ ∗ 2∧ 8 16 32
Proof. This follows from [KL02][KLW04] and [RW80]. (cid:3)
We want to be more specific in choice of the classes a . Let
i
R∗ :E2∗BString ∼=E2∗BSpin[[r˜]]−→E2∗BSpin
be the ring homomorphism given by the constant coefficient in the power series
expansion. Then dually, we get a map
R :E BSpin E BString.
2 2
∗ ∗ −→ ∗
Lemma 2.7. Let a be the image of b under this map. Then we state the equality
i i
a ,c = b ,R (c)
i i ∗
h i h i
for later purposes.
3. The action of the Morava stabilizer group
Inthissectionweanalyzethe actionofthe MoravastabilizergrouponE-theory.
We denote the formal group law obtained from the coordinate t= x on Cˆ by F,
−y
then elements of S =Aut(F) are certain power series g(t) F [[t]].
2 4
∈
3.1. The action on the formal group for Morava E-theory. Onecanlift the
curve C to
C :y2+3u xy+(u3 1)y =x3
U 1 1−
over (E ) =W(F )[[u ]], whose formal group law F is the universaldeformation
2 0 4 1 U
of F.
An automorphism of F (i.e. a power series g over F ) can be lifted to a power
4
series g˜ with coefficients in W(F ). Then
4
(Fg˜)(x,y)=g˜ 1(F (g˜(x),g˜(y))).
U − U
This is isomorphic, but in general not -isomorphic to F . It is a deformation of
U
∗
F, so it is classified by a ring homomorphism
hg :(E ) =W(F )[[u ]] (E ) =W(F )[[u ]],
2 0 4 1 2 0 4 1
→
which is characterizedby the property that (hg) F is -isomorphic to Fg˜.
∗ U ∗ U
The composition g (E ) [[z]] of the isomorphism of formal group laws
U 2
∈ ∗
g˜:F Fg˜
U → U
with the unique -isomorphism
∗
Fg˜ (hg) F
U → ∗ U
is the unique lift g of g such that FgU canbe obtained by pushing forwardF by
U U U
a ring isomorphism, namely hg.
8 GERDLAURESANDMARTINOLBERMANN
It follows that for each g S we have a commutative square
2
∈
Cˆ gU // Cˆ
U U
(cid:15)(cid:15) hg (cid:15)(cid:15)
Spec (E ) //Spec (E )
2 0 2 0
We denote
g (z)=t (g)z+t (g)z2+t (g)z3+...
U 0 1 2
3.2. Theaction onMoravaE-theory. Wearenowpreparedtodefinetheaction
on the coefficients (E ) . We extend hg :(E ) (E ) by
2 2 0 2 0
∗ →
hg(u)=t (g)u
0
to an automorphism hg of (E ) . More generally, we define the S -action on
2 2
∗
(E ) X =(E ) MU X
2 ∗ 2 ∗⊗MU∗ ∗
for a spectrum X [Rez98, Sect. 6].
We obtain a graded formal group law F over (E ) by setting
U 2
∗
F (x,y)=u 1F (ux,uy).
U − U
Theisomorphismg :F (hg) F inducesanisomorphismg :F (hg) F
U U → ∗ U U U → ∗ U
of graded formal group laws via
gU(z)=(hg(u))−1gU(uz).
This is classified by a ring homomorphism φ : MU MU (E ) sending t
2 i
MU MU to the (i+1)-st coefficient t (g)t (g) 1ui ∗(E ) → of g∗. ∈
T∗hemapφη :MU E classifiesiF a0ndφ−η :∈MU2 2−2Ei claUssifies(hg) F .
L 2 U R 2 U
∗ → ∗ → ∗
Using the MU MU-coaction
∗
ψ :MU X MU MU MU X,
∗ → ∗ ⊗MU∗ ∗
the action of g on (E ) X is now given as
2
∗
(E2)∗ φηL ⊗MU∗ MU∗X h−g→⊗ψ (E2)∗ φηR ⊗φMηUR∗ MU∗MU ⊗MU∗ MU∗X
1⊗−φ→⊗1(E2)∗ φηR ⊗φMηUR∗ (E2)∗ φηL ⊗MU∗ MU∗X −m→·1 (E2)∗ φηL ⊗MU∗ MU∗X.
We have defined, for each g S , a natural transformation (E ) ( ) (E ) ( )
2 2 2
∈ ψg ∗ − → ∗ −
ofhomologytheories,andthereforea mapofspectra E E up to homotopy(as
2 2
→
there are no phantom maps).
Vice versathe actionofthe stabilizergrouponE -homologyisgivenby sending
2
ahomologyclassx (E2) X,i.e. S x E2 X toψg(x)definedasS x E2 X ψg∧1
∈ ∗ → ∧ → ∧ →
E X, and the action on a cohomology class X E is given by composition
2 2
∧ →
with ψg :E E . This has the property that with the KroneckerpairingE X
2 → 2 2∗ ×
(E ) X (E ) we have
2 2
∗ → ∗ ψg b,x = ψgb,ψgx .
h i h i
CANNIBALISTIC CLASSES OF STRING BUNDLES 9
3.3. The action on the E-Euler class. The action of S = Aut(Cˆ) on the
2
complex coordinate x (E )2CP is given by
2 ∞
∈
g x=g (x)= t (g)t (g) 1uixi+1.
· U i 0 −
Xi 0
≥
For z =ux (E )0CP we have g z =g (z).
2 ∞ U
∈ ·
Beaudry shows in [Bea15] that
2t (g)
1
g u =t (g)u + .
1 0 1
· 3t (g)
0
The element ω corresponds to the power series g(t) = αt, which one can lift to
g˜(t)=ωt. The automorphism hω defined by
ω u =ωu
1 1
·
satisfies (hω) F˜ = F˜g˜, i.e. the push-forward is in this case not only -isomorphic,
but equal to∗F˜g˜. In particular we have ∗
ω u=ωu,
·
ω x=x.
·
Strickland uses the compatibility of the GL (Z/3)-actionon T (CP ) with the
2 ∗ ∞
G -action on E CP to prove that [Bea15, Section 2.4]
2 2∗ ∞
1 2ω
i u=u− − ,
· u 1
1
−
u +2
1
i u = .
1
· u 1
1
−
(Note that z z+2 is the automorphism of P1 sending 1 , 1,ω
7→ z 1 7→ ∞ ∞ 7→ 7→
ω2,ω2 ω.) −
7→
Moreover we have
lz+rlw(z)
i z = ,
· 1+sz+l3(sr t)w(z)
−
where
1 2ω 1 u3 ω2u 1 u3 1
l= − − , r =3 − 1 , s=3 1− , t=3 1− 1 ω+(1 ω2)u
u 1 (u 1)3 u 1 (u 1)4 − − 1
1− 1− 1− 1− (cid:0) (cid:1)
and w(z) (E ) [[z]] is the power series with leading term z3 given by solving
2
∈ ∗
w+3u zw+(u3 1)w2 =z3
1 1−
for w.
The actionof the Frobenius is trivial on u,u , but is the conjugation on W(F ).
1 4
3.4. The action on the E-homology of K(Z,3). After the description of the
operation on a complex coordinate in E K(Z,2), we can address the more com-
2∗
plicated operation on the generator of E K(Z,3). The following result is related
2∗
to [Pet] and [Wes, Theorem 3.21]where the analogousformula for the homology is
proven.
Theorem 3.1. The S -action on E K(Z,3)=E [[r˜ ]] is given by
2 2∗ 2∗ K
ψg(r )=rdet(g)
K K
10 GERDLAURESANDMARTINOLBERMANN
Proof. Theclassr ismultiplicativeforsumsofvectorbundles. Hence,theH-space
U
addition :BU 6 2 BU 6 sends
⊕ h i → h i
rU 7→rU ⊗rU ∈E2∗BUh6i⊗E2∗ E2∗BUh6i.
Since j : K(Z,3) BU 6 is an H-space map which maps r r , the H-space
U K
→ h i 7→
addition :K(Z,3)2 K(Z,3) sends
⊕ →
rK 7→rK ⊗rK ∈E2∗K(Z,3)⊗E2∗ E2∗K(Z,3).
Let ψg(r )=q(r˜ ) for q E [[t]] a power series with leading term 1. Then
K K ∈ 2∗
q(r˜K) q(r˜K)=ψg(rK) ψg(rK)=ψg( ∗(rK))= ∗ψg(rK)
⊗ ⊗ ⊕ ⊕
= (q(r˜ ))=q( (r˜ ))=q(r˜ 1+1 r˜ +r˜ r˜ )
∗ K ∗ K K K K K
⊕ ⊕ ⊗ ⊗ ⊗
Thus q(t) is a power series with leading term 1 satisfying
(3.1) q(t )q(t )=q(t +t +t t ).
1 2 1 2 1 2
Forα E therecannotexistmorethanonepowerserieswithleadingterms1+αt
∈ 2∗
satisfyingtheequation(useinductionandcomparethedegreencoefficientsonboth
sides of the equation to see that the degree n term of the series is unique, as the
groundringistorsionfree). Thisistrueevenifweinvert2,whenallbinomialseries
B (t)= α tn forallα E satisfythisequation. Itfollowsthatψg(r )=rf(g)
α n ∈ 2∗ K K
for somePexp(cid:0)o(cid:1)nent f(g), where the power is defined by the binomial series.
We claim that f already maps to Z×2. For α = f(g) we know that all binomial
coefficients α are integral. It follows for n = p that α coincides with an integer
n
k Z mod p(cid:0)E(cid:1). Similarly, for n=p2 one obtains that α coincides with an integer
∈ 2∗
modulo p2 and so on.
Since r is multiplicative we obtain a homomorphism
K
f :G2 −→Z×2.
It remains to show that this homomorphism is the determinant.
Consider the product
β
µ :K(Z/n,1) K(Z/n,1) K(Z/n,2) K(Z,3)
n
× −→ →
which is derived from the product structure of the Eilenberg-MacLane spec-
trum and the Bockstein map. It is additive in each of the two arguments, and
spf(E0(K(Z,3))is isomorphicto themultiplicative formalgroup,cf. [Wes,Propo-
2
sition 3.2]. In fact, this map induces the universal e -pairing in the sense of Ando
n
andStrickland[AS01,Proposition2.3]: writeG[n]forthe theformalgroupscheme
spf(E0(K(Z/n,1))), then µ induces a pairing
2 n
f :G[n] G[n] spf(E0(K(Z,3)))
× −→ 2
with the properties
f(x +x ,y) = f(x ,y)f(x ,y),
1 2 1 2
f(x,y +y ) = f(x,y )f(x,y ),
1 2 1 2
f(x,x) = 1.
(Here we use the language of R-points for R an E0-algebra.) The left action of
2
the Morava stabilizer group on the cohomology induces a right action on formal
schemes via precomposition. Writing g instead of ψg we have the identity
(3.2) f(xg,yg) = f(x,y)g.
