The Grothendieck--Teichmüller group and the stable symplectic category

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by  Nitu Kitchloo

THE GROTHENDIECK–TEICHMU¨LLER GROUP AND THE STABLE
SYMPLECTIC CATEGORY
NITUKITCHLOOANDJACKMORAVA
3 ABSTRACT. We consider an oriented version of the stable symplectic categorydefined in
1 [27]. We define a canonical representation (or fiber functor) on this category, and study
0 its motivic group of monoidal automorphisms. In particular, we observe that this Galois
2 groupcontainsanaturalsubgroupisomorphictotheabelianquotientoftheGrothendieck–

n Teichmu¨llergroup. WealsostudytherationalWaldhausenK-theoryoftheE∞-ringspec-
a trum of coefficientsΩ of the stable symplectic category, and itsrelationtothe symplecto-
J
morphismgroupofanobjectinthestablesymplecticcategory.

1
3

]
T
A CONTENTS
.
h
1. Introduction 1
t
a
m 2. The Stable Symplectic homotopy category 2
[ 3. The WaldhausenK-theory of sΩand aSymplectic invariant. 6

2 4. Acanonical representation 8
v
5 5. Appendix: Grothendieck-Teichmu¨ller groups 14
0
9 References 18
6
.
2
1
2
1 1. INTRODUCTION
:
v
i In [27] the first author defined a stabilization hS of the symplectic category introduced
X
by A. Weinstein in [36, 38]. The objects of Weinstein’scategory are symplectic manifolds,
r
a and the morphisms between two symplectic manifolds (M,ω) and (N,η) are lagrangian
immersions into M × N, where the conjugate symplectic manifold M is defined by the
pair (M,−ω). The composition L ∗ L of two lagrangian immersions L # M ×N and
1 2 1
L # N × K, is defined to be the fiber product: L × L −→ M × K. This definition
2 1 N 2
doesnotalwaysyieldalagrangianimmersiontoM×K: todoso,thepullbackdefiningit
must betransverse, so Weinstein’sconstruction isunfortunately not a genuinecategory.
In [27], we described a way to extend the symplectic category to an honest category hS,
byintroducing amodulispaceofstabilized(inthesenseofhomotopytheory)lagrangian
immersions in a symplectic manifold of the form M ×N 1. This moduli space can be de-
scribed as the infinite loop space corresponding to a certain Thom spectrum. Taking this
Date:February1,2013.
NituKitchlooissupportedinpartbyNSFthroughgrantDMS1005391.
1undertheassumptionofmonotonicity. Otherwise,onehasthespaceoftotallyrealimmersions.
1
as the space of morphisms defines a stable symplectic (homotopy) category hS that is natu-
rallyenrichedoverthehomotopycategoryofspectra(undersmashproduct)2. Composi-
tioninhSiswell-definedandremainsfaithfultoWeinstein’soriginaldefinition. Geomet-
rically, the stabilization of Weinstein’s category can be seen as “inverting the symplectic
manifold C”, analogous to the introduction of an inverse to the projective line in the the-
ory of motives. In other words, we introduce a relation on the symplectic category that
identifiestwosymplecticmanifoldsM andN ifM×Ck becomesequivalenttoN×Ck for
some k. This stable symplectic category has variants defined by lagrangian immersions
with oriented andmetaplectic structures.
Herewe studya monoidal functor from aclosely related stableoriented symplectic cate-
gorysStothecategoryofmodulesoveranaturallyasociatedringspectrum sΩdescribed
as a Thom spectrum (U/SO)−ζ. On a symplectic manifold M the value of this functor is
an sΩ-module sΩ(M) representing the space of stably immersed oriented lagrangians in
M. By extending coefficients to other algebras over sΩ, one has a family of such func-
tors, and we can ask for the structure of the Galois group of monoidal automorphisms
of this family. We answer this question (see corollary 4.6 and theorem 4.9), and relate
this motivic group to the Gothendieck–Teichmu¨ller group [30]. We also study the ratio-
nalWaldhausenK-theory ofsΩ; inparticular, weconstruct aninteresting invariantofthe
classifying space of the symplectomorphism group of an object (M,ω) with values in the
WaldhausenK-theory ofsΩ.
Thisdocumentisorganizedasfollows: Insection2werecalltheconstructionofthestable
symplectic homotopy category. Section 3 describes a computation of the rational Wald-
hausen K-theory of the ring spectrum sΩ, with applications to the symplectomorphism
group. Insection4wedescribethealgebraicrepresentationsthatweareinterestedin,and
identifytheGaloisgroupofmonoidalautomorphismsofitsrationalization. Section4also
describes the construction of an integral model for this Galois group. Section 5 summa-
rizes some properties of the Grothendieck-Teichmu¨ller group and related constructions,
and applies ideas from Koszul duality (over Q). In particular, we define a Hopf-Galois
analog ofthe Grothendieck-Teichmu¨ller group in the category of spectra, and identifyits
abelianization, overQ, with the group of symmetries identified in Section 4.
Before we begin, we would like to thank M. Abouzaid, V. Angelveit, A. Baker, Y. Eliash-
berg, D. Gepner, M. Hazewinkel, J. Lind, B. Richter and H. L. Tanaka for helpful conver-
sations related to thisproject.
2. THE STABLE SYMPLECTIC HOMOTOPY CATEGORY
Inthissectionwerecalltheconstruction ofthestablesymplectichomotopycategory[27].
Givenasymplecticmanifold(M,ω)ofrealdimension2m,weconstructaspectrumΩ(M)
so that the corresponding infinite loop space can be interpreted as a space whose points
represent manifolds that admit totally-real immersions into M × Cn for large values of
n (up to an equivalence that shall be made precise later). We will say that M satisfies
monotonicity if the cohomology class of the symplectic form ω is a scalar multiple of the
first Chern class of M. Under the assumption of monotonicity, totally real immersions
maybe replacedby lagrangian immersions in the above interpretation ofΩ(M).
2In[27],weliftedthiscategorytoanA∞-categoryenrichedoverspectra.
2
§2.1 The basicconstruction
ConsidertheThomspectrumΣnG(τ ⊕Cn)−ζn,wherethebundleζ isdefinedbyvirtueof
n
the pullbackdiagram:
G(τ ⊕Cn) ζn // BO(m+n)
(cid:15)(cid:15) (cid:15)(cid:15)
M τ⊕Cn // BU(m+n),
where τ denotes(homotopy unique)complexstructure on thetangent bundleofM com-
patiblewiththesymplecticform ω. In[27],weusedtheworkofD.Ayala[4]toshowthat
for n > 0, the infinite loop space Ω∞−n(G(τ ⊕ Cn)−ζn) can be interpreted as the moduli
space of manifolds Lm+n ⊂ R∞ × Rn, with a proper projection onto Rn, and endowed
with a totally-real immersion Lm+k # M × Cn (or lagrangian immersion, under the as-
sumption of monotonicity). More precisely, the space Ω∞−n(G(τ ⊕ Cn)−ζn) is uniquely
defined by the property that given a smooth manifold X, the set of homotopy classes of
maps [X,Ω∞−n(G(τ ⊕ Cn)−ζn)], is in bijection with concordance classes of smooth man-
ifolds E ⊂ X × R∞ × Rn over X, so that the first factor projection: π : E −→ X is a
submersion, and which are endowed with a smooth map ϕ : E −→ M × Cn which re-
stricts to a totally-real immersion (resp. lagrangian) on each fiber of π. As before, we
demandthat the third factor projection E −→ Rn befiberwise properover X.
Nowthe standard inclusion Rn1 ⊆ Rn2, inducesa natural map:
ϕn1,n2 : Σn1G(τ ⊕Cn1)−ζn1 −→ Σn2G(τ ⊕Cn2)−ζn2,
which represents the map that sends a concordance class E, to E × Rn2−n1, by simply
taking the product with the orthogonal complementof Rn1 in Rn2.
Definition2.1. DefinetheThomspectrumΩ(M)representingtheinfiniteloopspaceofstabilized
totally-real (resp. lagrangian under the assumption of monotonicity) immersions in M to be the
colimit:
Ω(M) = G(M)−ζ := colim ΣnG(τ ⊕Cn)−ζn.
n
Notice that by definition, we have a canonical homotopy equivalence: Ω(M × C) ≃ Σ−1Ω(M).
Taking M to be a point, we define Ω = Ω(∗) = (U/O)−ζ, where the bundle ζ over U/O is the
virtual zerodimensionalbundle over(U/O) definedby thecanonical inclusionU/O −→ BO.
Henceforth, we shall use the term “lagrangian immersion” to mean “totally-real immer-
sion” if the condition of monotonicity fails to hold. We take this opportunity to also
introduce the (abusive) convention of not decorating the stable vector bundle ζ by the
underlying manifold M. Hopefully, the manifold M will be clearfrom context.
We may also describe Ω(M) as a Thom spectrum: Let the stable tangent bundle of M of
virtual (complex) dimension m be given by a map τ : M −→ Z × BU. As the notation
suggests, letG(M) be definedasthe pullback:
ζ
G(M) // Z×BO
(cid:15)(cid:15) (cid:15)(cid:15)
M τ // Z×BU.
3
Then the spectrum Ω(M) is homotopy equivalent to the Thom spectrum of the stable
vector bundle −ζ overG(M) definedin the diagram above.
Notice that the fibration Z × BO −→ Z × BU is a principal bundle up to homotopy,
with fiber being the infinite loop space U/O. Hence, the spectrum Ω(M) is homotopy
equivalent to a Ω-module spectrum. Now, observe that we have the equivalence, up to
homotopy, ofU/O-spaces:
G(M)× G(N) ≃ G(M ×N).
U/O
Thistranslates to a canonical homotopy equivalence:
µ : Ω(M)∧ Ω(N) ≃ Ω(M ×N).
Ω
Let us now describe the stable symplectic homotopy category hS. The objects of this cat-
egory will be symplectic manifolds (M,ω) (see remark 2.3), endowed with a compatible
almost complexstructure.
Definition 2.2. The spectrumΩ(M,N) of morphismsin hS from M to N is the Ω-module spec-
trum:
Ω(M,N) := Ω(M ×N).
Remark 2.3. Notice that objects in hS need not be compact. The price we pay for this, familiar
fromother contexts, isthatwe simplylosethe identitymorphismsfor non-compactobjects.
The nextstep is todefine composition. Thesimplest case
Ω(M,∗)∧ Ω(∗,N) −→ Ω(M,N),
Ω
is the map µ constructed earlier. For the general case, consider k + 1 objects objects M
i
with 0 ≤ i ≤ k, and letthe space G(∆) be definedbythe pullback:
G(∆) // G(M ×M ×···×M ×M )
0 1 k−1 k
ξ
(cid:15)(cid:15) (cid:15)(cid:15)
M ×(M ×···×M )×M ∆ // M ×(M ×M )×···×(M ×M )×M
0 1 k−1 k 0 1 1 k−1 k−1 k
where ∆denotesthe product to the diagonals∆ : M −→ M ×M ,for 0 < i < k.
i i i
Now notice that the fibrations defining the pullback above are direct limits of smooth
fibrations with compact fiber. Furthermore, the map ∆ is a proper map for any choice
of k + 1-objects (even if they are non-compact). In particular, we may construct the
Pontrjagin–Thom collapse map along the top horizontal map by defining it as a direct
limitof Pontrjagin–Thom collapsesfor each stage.
Let ζ denote the individual structure maps G(M × M ) −→ Z × BO, and let η(∆)
i i−1 i
denote the normal bundle of ∆. Performing the Pontrjagin–Thom construction along the
top horizontal mapin the above diagram yieldsamorphism of spectra:
ϕ : Ω(M ,M )∧ ···∧ Ω(M ,M ) ≃ Ω(M ×M ×···×M ×M ) −→ G(∆)−λ
0 1 Ω Ω k−1 k 0 1 k−1 k
where λ : G(∆) −→ Z×BO is the formal difference of the bundle ζ and the pullback
L i
bundle ξ∗η(∆).
The next step in defining composition is to show that G(∆)−λ is canonically homotopy
equivalent to Ω(M ,M ) ∧ (M × ··· × M ) , where (M × ··· × M ) denotes the
0 k 1 k−1 + 1 k−1 +
4
manifold M × ··· × M with a disjoint basepoint. To achieve this, it is sufficient to
1 k−1
construct aU/O-equivariantmap over M ×(M ×···×M )×M :
0 1 k−1 k
ψ : G(M ×M )×(M ×···×M ) −→ G(∆),
0 k 1 k−1
thatpullsλbacktothebundleζ ×0. Theconstruction ofψ isstraightforward. Wedefine:
ψ(λ,m ,...,m ) = λ⊕∆(T (M ))⊕···⊕∆(T (M )),
1 k−1 m1 1 mk−1 k−1
where ∆(T (M)) ⊂ T (M × M) denotes the diagonal lagrangian subspace. Now let
m (m,m)
π : G(∆)−λ −→ Ω(M ,M )betheprojection mapthatcollapsesM ×···×M toapoint.
0 k 1 k−1
Definition 2.4. We definethe compositionmapto be theinducedcomposite:
πϕ : Ω(M ,M )∧ ···∧ Ω(M ,M ) −→ G(∆)−λ −→ Ω(M ,M ).
0 1 Ω Ω k−1 k 0 k
We leaveittothe readertocheckthatcompositionas definedabove is homotopyassociative.
§2.2 The identitymorphism:
Wenowshow that acompact manifold (M,ω) hasanidentity morphism:
Proposition 2.5. LetM beacompactmanifold,and let[id] : S −→ Ω(M,M)bearepresentative
ofhomotopyclassofthediagonal(lagrangian)embedding∆ : M −→ M×M. Then[id]isindeed
the identityfor thecompositiondefined above.
Proof. GiventwomanifoldsM,N,let∆(M) ⊂ M ×M beadiagonalrepresentative of[id]
asabove. ObservethatN×∆(M)×M istransversetoN×M×∆(M)insideN×M×M×M.
TheyintersectalongN×∆ (M),where∆ (M) ⊂ M×M×M isthetriple(thin)diagonal.
3 3
Hencewe get adiagram
Ω(N,M)∧S // Ω(N,M)∧∆(M)−τ ∆−τ// Ω(N,M)∧ Ω(M,M)
SSSSSSSSSSSSSS)) (cid:15)(cid:15) Ω(cid:15)(cid:15)
Ω(N,M) = // Ω(N,M)
commutative up to homotopy, where the right vertical map is composition, and the left
vertical mapisthe Pontrjagin–Thom collapse overthe inclusion map
N ×M = N ×∆ (M) −→ N ×M ×∆(M).
3
Nowconsider the following factorization of the identitymap:
N ×M = N ×∆ (M) −→ N ×M ×∆(M) −→ N ×M
3
wherethelastmapistheprojection ontothefirsttwofactors. PerformingthePontrjagin–
Thom collapse over this composite shows that
Ω(N,M)∧S −→ Ω(N,M)∧∆(M)−τ −→ Ω(N,M).
is the identity. It follows that right multiplication by [id] : S → Ω(M,M) induces the
identitymap onΩ(N,M), upto homotopy. Asimilarargumentworks for leftmultiplica-
tion. (cid:3)
5
Remark2.6. RecallthatgivenarbitrarysymplecticmanifoldsM and N, thecompositionmap:
Ω(M,∗)∧ Ω(∗,N) ≃ Ω(M,N)
Ω
induces a natural decompositionof Ω(M,N). In particular, arbitrary compositions canbe canon-
icallyfactoredusing thisdecomposition,and canbe computedby applyingthe “inner product”
Ω(∗,N)∧ Ω(N,∗) −→ Ω.
Ω
tothe factors.
Itwill also beimportant belowthat hSis asymmetric-monoidal category, with a product
given by the cartesian product ofsymplectic manifolds.
Definition2.7. TheconstructionoftheorientedstablesymplectichomotopycategorysSiscom-
pletely analogous, but with O replaced by SO; the commutative ring spectrum sΩ = (U/SO)−ζ
definesits coefficients.
NowΩisan(Eilenberg–MacLane)H(Z/2)-algebra, sowe canregard Sasacategory with
morphismobjectsenrichedoveraclassicaldifferentialgradedalgebra. Thisisnotthecase
forsΩ,butitsrationalizationsΩ⊗QisagainageneralizedEilenberg–MacLanespectrum,
with
sΩ ⊗Q = Λ [y , i > 0]
∗ Q 4i+1
an exterior algebra on certain odd - degree generators. Moreover, the category sS simpli-
fiesconsiderably when rationalized. Inparticular, the Thom isomorphism
sΩ(M) ⊗Q ∼= H (M,sΩ ⊗Q)
∗ ∗ ∗
identifies sS⊗Q with an (Arnol’d-Ho¨rmander-Maslov...) category of symplectic mani-
folds, whose morphisms areclassical cohomological correspondences with compact sup-
port, butwith coefficients in the graded ring sΩ ⊗Q ∼= H (U/SO ,Q).
∗ ∗ +
3. THE WALDHAUSEN K-THEORY OF sΩ AND A SYMPLECTIC INVARIANT.
InthissectionwewillstudythecoefficientspectrumsΩthroughitsWaldhausenK-theory.
Thespectrum sΩisaconnective spectrum with π (sΩ) = Z. Letusconsiderthe fibration:
0
K(π) −→ K(sΩ) −→ K(Z)
where π isthe fiberof the zero-th Postnikov section: sΩ → HZ.
Proposition 3.1. Let K(sΩ) denote the cofiber of the canonical map K(S) → K(sΩ). Then ra-
tionally, the spectrum K(π) is equivalent to K(sΩ). In particular the above fibration admits a
canonical rational splitting, and there exist polynomial classes y in degree 4i + 2 such that
4i+2
π K(π) isisomorphicto theaugmentation ideal:
∗
π K(π)⊗Q = Q[y ] .
∗ 4i+2 >0
Furthermore, rationally π K(π) can be identified with the (injective) image in homotopy of the
∗
group completionmap:
Ω∞Σ∞(B(U/SO)) −→ BGl (sΩ)+ = Ω∞K(sΩ).
∞
6
Proof. SinceK(S)isrationallyequivalenttoK(Z),thefirstpartoftheclaiminclear. Viathe
Thom isomorphism, we may identify sΩ rationally with the ring spectrum Σ∞(U/SO) .
+
In particular π sΩ ⊗ Q = Λ(y ). Now invoking results from [3], we see that π K(π)
∗ 4i+1 ∗
can be identified with the set of positive degree elements in THH (U/SO ) in the kernel
∗ +
of the Connes boundary operator. These elements are given by the augmentation ideal
in the polynomial algebra H (B(U/SO)) = Q[y ]. The classes y (ancestors of the
∗ 4i+2 4i+2
exterior classes y ) are detected in rational homotopy along the inclusion given by:
4i+1
B(U/SO) −→ BGl (Σ∞(U/SO) ))+. The result nowfollows. (cid:3)
∞ +
We now fix a compact symplectic manifold (M,ω), and describe an invariant of the clas-
sifyingspaceofthesymplectomorphism group: BSymp(M), withvaluesinK(sΩ). Recall
from [27]that one hasa map:
γ : BSymp(M) −→ BGl(sΩ(M,M)),
where the group of units GL(sΩ(M,M)) = Aut (sΩ(M)) is defined as the components
sΩ
that induce invertible π (sΩ)-module maps in homotopy. The map γ is defined by de-
0
looping the map that sends a symplectomorphism ϕ to its graph in M ×M. Recall [27],
that the map γ was explicitly constructed as a map that classifies a parametrized bundle
of sΩ-module spectra over BSymp(M) with fiber sΩ(M). This bundle was obtained as a
fiberwise compactification of the formal negative of a bundle J(ζ) defined over a space
G(J(M)) fiberingover BSymp(M):
J(ζ)
BSymp(M) oo G(J(M)) // Z×BO
= ξ
(cid:15)(cid:15) (cid:15)(cid:15) J(τ) (cid:15)(cid:15)
BSymp(M) oo J(M) // Z×BU,
where J(M) −→ BSymp(M) is equivalent to the universal fiber bundle with fiber M and
structure group Symp(M).
Definition 3.2. Since sΩ(M) isafinite cellularsΩ-module,we maydefine astabilizationmap:
Q(M) : BGl(sΩ(M,M)) −→ Ω∞K(sΩ)
whereK(sΩ)denotesWaldhausen’sK-theoryspectrumofthecommutativeringspectrumsΩ. The
K(sΩ)-valued parametrizedindex isdefinedas thecomposite:
I(M) = Q(M)◦γ : BSymp(M) −→ Ω∞K(sΩ).
Note thatsuchastabilization canbe definedforthe categorySas well.
Theorem 3.3. In cohomology, the parametrized index map I(M) can be identified with the map
given by applying the Becker–Gottlieb transfer: [tr] : H∗(J(M)) −→ H∗(BSymp(M)) to a sub-
algebra inside H∗(J(M)). This sub algebra is generated by the odd Newton polynomials (see dis-
cussionbefore4.5)intheChernclassesof thefiberwisetangentbundle J(M) −→ BSymp(M).
Proof. By [21](Thm 8.5),the map I(M) hasa factorization:
BSymp(M) −→ Ω∞Σ∞(J(M) ) −→ Ω∞A(J(M)) −→ Ω∞K(sΩ)
+
where A(J(M)) is the Waldhausen K-theory of J(M), the first map is the Becker–Gottlieb
transfer, the second is the natural transformation from Σ∞(X ) to A(X), and the final
+
map isthe one thatclassfies the stable bundle of sΩ-spectra overJ(M).
7
Now let BGl(sΩ) denote the classifying space of the group of units of sΩ [5], which clas-
sifies bundlesof parametrized rank one sΩ-module spectra. Since sΩ is an E -ring spec-
∞
trum, BGl(sΩ) is a an infinite loop space, with product defined by the fiberwise smash
product of parametrized sΩ-module spectra.
The projection map U −→ U/SO is equivalent to a map of infinite loop spaces, and the
bundle ζ over U/SO (that defines sΩ = (U/SO)−ζ as a Thom spectrum) restricts to the
trivialbundleoverU. Conseuentlywehavealeft-actionmap: U ∧(U/O)−ζ −→ (U/O)−ζ.
+
Thistranslates to a mapof H-spacesBU −→ BGl(Ω).
Using naturality, we have the following description of I(M) given by the commutative
diagram with the vertical maps induced by the map that classifies the (rank one) bundle
ofsΩ-spectra overJ(M):
Σ∞(BSymp(M) ) tr // Σ∞(J(M) ) // A(J(M)) // K(sΩ)
+ +
= =
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
Σ∞(BSymp(M) ) // Σ∞(BU ) // A(BU) // K(sΩ)
+ +
= =
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
Σ∞(BSymp(M) ) // Σ∞(BGl(sΩ) ) // A(BGl(sΩ)) // K(sΩ)
+ +
Now recall from the proof of claim 3.1, that the map BU −→ Ω∞K(sΩ) is surjective onto
the vector space spanned by the generators y in rational homotopy. It follows that
4k+2
in rational cohomology, the map: H∗(Ω∞K(sΩ)) −→ H∗(BU) is surjective onto the poly-
nomial algebra generated by the odd Newton polynomials N (c ,c ,...,c ) in the
2k+1 1 2 2k+1
universal Chern classes. Invoking the above diagram, the proof follows. (cid:3)
Remark3.4. Proposition3.1showsthat wehave arational pullbackdiagram:
Σ∞B(U/SO) // S
+
(cid:15)(cid:15) (cid:15)(cid:15)
K(sΩ) // K(Z)
Hence we may define secondary rational (Reidemeister) invariants in dimensions 4k, for families
of symplecticmanifolds(M,ω)thatadmitaprescribednull homotopyfor theparametrizedindex.
4. A CANONICAL REPRESENTATION
The next item on our agenda is a canonical representation of hS. For the applications
we have in mind, hsS will be the relevant category, but for the moment we proceed with
constructions which work generally.
Given a symplectic manifold (M,ω) recall that the morphism spectrum Ω(∗,M) can be
identified with the Ω-module spectrum Ω(M). In particular, right composition in hS
yieldsa representation:
F : hS −→ ΩS, F(M) = Ω(M),
8
where ΩS denotes the homotopy category of Ω-module spectra. Recall that the category
hS isa symmetric-monoidal category, with the monoidal structure given by the cartesian
product of symplectic manifolds. Since Ω(M × N) is equivalent to Ω(M) ∧ Ω(N), the
Ω
functorF ismonoidal. ThisfunctorhasamotivicgroupAut (Ω)ofnaturalA Ω-module
∧ ∞
equivalences with itself. This maps to a group G of monoidal automorphisms of the
Ω
functor
F : E 7−→ [M 7→ Ω(M)∧ E]
E Ω
on the category of Ω-module spectra. The analogous construction in hsS is nontrivial
whentensoredwithQ,whichhastheadvantageofallowingustoworkwithLiealgebras
rather than the groups themselves.
§4.1 The Lie algebraof primitive automorphisms:
Given a symplectic manifold M, recall that Ω(M) was a defined as a Thom spectrum:
G(M)−ζ. Here π : G(M) −→ M was a principal U/O-bundle, supporting a stable real
vector bundle ζ : G(M) −→ BO of virtual dimension m. The bundle π is classified by the
map:
τ(Ω,M) : M −→ B(U/O).
Theorem 4.1. Given an objectof hS represented by a symplecticmanifold (M,ω), then τ(M,Ω)
factors through the map τ(M) : M −→ BU that classfies the tangent bundle of M, followed by
the projection map BU −→ B(U/O). Furthermore, the restriction of τ(Ω,M × M) along the
diagonal: ∆ : M −→ M ×M istrivial.
Proof. Notice that the bundle π : G(M) −→ M was induced from the frame bundle of
M, with structure group U, along the left action of U on U/O, so we can lift the map
τ(Ω,M), to BU, to get the factorization through τ(M). This proves the first part of the
claim. Nownotice thattherestriction ofthetangentbundleofM ×M alongthediagonal
∆iscanonicallyisomorphictothecomplexificationofthetangentbundleofM. Thismay
be restated assaying that one hasa uniquelift τ(∆) that makesthe following commute:
τ(∆)
M // BO
∆
(cid:15)(cid:15) (cid:15)(cid:15)
M ×M τ // BU.
Itfollows that τ(Ω,M ×M) istrivial when restricted along∆. (cid:3)
Remark 4.2. The projection map BU −→ B(U/O) is injective in cohomology away from two.
It is easy to see that its image is generated by classes d in degree 4i + 2, whose generating
2i+1
functioncan beexpressedintermsof the Chernclassesas:
(−1)ic
Xdi = Pi i ≡ 1−X2c2i+1 +decomposables.
c
i Pi i i
Similarly,ifoneconsiderstheinclusionBO −→ BU,thenthismapisinjectiveinhomologyaway
fromtwo. Ifwelet b denotethehomogeneousgeneratorsintheimageofH (CP∞) ⊂ H (BU),
Pi i ∗ ∗
thentheimageofH (BO)isthesub(polynomial)algebrageneratedbyclassesa (awayfromtwo),
∗ i
givenby:
Xai = (Xbi)(X(−1)ibi) ≡ 1+X2b2i +decomposables.
i i i i>0
9
Definition 4.3. Henceforth, we work in the oriented category hsS, and we assume that π E is a
∗
Q-vectorspace. DefineP (B(U/SO))tobethegradedE∗-submoduleofE˜∗(B(U/SO))consisting
E
of primitiveelementsinthe (commutative)HopfalgebraE∗(B(U/SO)).
Theorem 4.4. P (B(U/SO)) acts on F by graded natural transformations. In other words,
E E
thereis anatural mapof gradedE∗-modules:
P(E) : P (B(U/SO)) −→ End(F ).
E E
Furthermore,theimageofP(E)iscontainedinthesubgroupofprimitivenaturaltransformations,
definedas natural transformations ϕthat are additivewith respecttothe monoidalstructure:
ϕ(X ∧ Y) = ϕ(X)∧ Y +X ∧ ϕ(Y).
E E E
Proof. Fix an object (M,ω) of hsS. Given an element α ∈ P (B(U/SO)), we define the
E
action ofP(E)(α)on F (M) asthe cap product with α,described asthe composite map:
E
α : sΩ(M) −→ sΩ(M) ∧M −→ sΩ(M) ∧B(U/SO) −→ sΩ(M) ,
∗ E E + E + E
where the first map is induced by the diagonal map sG(M)−ζ −→ sG(M)−ζ ∧ M . The
+
second map is induced by τ(sΩ,M), and the third map above is given by capping with
the class α. [The reader should bear in mind that α = α(M) depends on M.] The
∗ ∗
construction above definesa mapof E∗-modules:
P(E) : P (B(U/SO)) −→ End (F (M)).
E E E
of E∗-modules. It remains to show that P(E) yields primitive natural transformations.
Consider symplectic manifolds M and N. Recall that sG(M × N) is equivalent to the
external product bundle sG(M) × sG(N). In particular, the element τ(sΩ,M × N)
U/SO
decomposes asthe composite:
M ×N −→ B(U/SO)×B(U/SO) −→ B(U/SO).
Given a primitive class α ∈ P (B(U/SO)), the pullback of α along τ(sΩ,M ×N) is there-
E
fore given by α(M) ∧ 1 + 1 ∧ α(N) . This is exactly the definition of a primitive endo-
∗ ∗
morphism.
Tosee thatP(E)(α) isa natural transformation, we needto showthat the diagram
sΩ(M) ∧ sΩ(M,N) α∗∧α∗// sΩ(M) ∧ sΩ(M,N)
E E E E E E
(cid:15)(cid:15) (cid:15)(cid:15)
sΩ(N) α∗ // sΩ(N) ,
E E
commutes; where the vertical maps are induced by composition in hS, and the top hor-
izontal map: α ∧ α : sΩ(M) ∧ sΩ(M,N) −→ sΩ(M) ∧ sΩ(M,N) denotes the
∗ ∗ E E E E E E
external smash product ofthe two mapsα(M) andα(M ×N) .
∗ ∗
Bythe primitivity ofα ,wewrite α(M ×N) asthe sumα(M) ∧1 + 1∧α(N) . Thisde-
∗ ∗ ∗ ∗
composition allowsustoreducethegeneralcasetothespecialcasewhenN isapoint. In
other words, we would like to show that the following specialcase of the above diagram
10