The diffeomorphism group of a K3 surface and Nielsen realization

0.23 MB·English

by  Jeffrey Giansiracusa

THE DIFFEOMORPHISM GROUP OF A K3 SURFACE AND NIELSEN
REALIZATION
JEFFREYGIANSIRACUSA
9
0 ABSTRACT. The Nielsen Realization problem asks when the group homomorphism
0 Diff(M) → π0Diff(M) admits a section. For M a closed surface, Kerckhoffproved
2
that a section exists over any finite subgroup, but Morita proved that if the genus is

n large enough then no section exists over the entire mapping class group. We prove
a the first nonexistence theorem of this type in dimension 4: if M is a smooth closed
J
oriented 4-manifold which contains a K3 surface as a connected summand then no
5
sectionexistsoverthewholeofthemappingclassgroup. Thisisdonebyshowingthat
1
certainobstructionslyingintherationalcohomologyofBπ0Diff(M)arenonzero. We

] detecttheseclasses by showingthattheyare nonzerowhenpulledbackto the moduli
T spaceofEinsteinmetricsonaK3surface.
G
.
h
t
a
m 1. INTRODUCTION
[

Let M be a smooth closed oriented manifold. We write Diff(M) for the group of ori-
2
v entationpreservingC∞ diffeomorphismsofM; thisisatopologicalgroupwiththeC∞
5 Fre´chet topology. The mapping class group π Diff(M) of M is the group of isotopy
4 0
classes ofdiffeomorphisms.
5
4
Asubgroupofthemappingclassgroupisrepresentedbyacollectionofdiffeomorphisms
.
5
which form a group up-to-isotopy. The (generalized) Nielsen realization problem asks
0
7 whenasubgroupofthemappingclassgroupofM canberectifiedtoanactualsubgroup
0 of the diffeomorphism of M. This is equivalent to asking if the projection Diff(M) →
:
v π Diff(M) admitsasectionoveragivensubgroupofthemappingclass group.
0
i
X
Inthecontextofsurfacesthisproblemhasalonghistory,originatingin[Nie43]. Kerck-
r
a hoff[Ker83]showedthatallfinitesubgroupsofthemappingclassgroupofasurfacecan
berectified. Morita[Mor87]thenshowedthattheMiller-Morita-Mumfordcharacteristic
classes κ ∈ H∗(BDiff(F );Q) ∼= H∗(Bπ Diff(F );Q) (i ≥ 2) provideobstructionsto
i g 0 g
rectifyinginfinitesubgroupsinthesmoothcase. Onthemappingclassgroupthefirst of
these obstructions is nonvanishingfor g ≥ 5. However, these classes no longer provide
obstructionsifonereplaces thediffeomorphismgroupwiththehomeomorphismgroup.
More recently, Markovic [Mar07] has shown that in the case of homeomorphisms no
sectionexistsovertheentiremappingclassgroup,again assumingg ≥ 5.
The purpose of this paper is to prove the first theorem deciding the Nielsen realization
problemforaclass of4-dimensionalmanifolds.
Date:November29,2008.
2000MathematicsSubjectClassification. 57R70;(14J28;58D27;57S05).
1
2 JEFFREYGIANSIRACUSA
Theorem 1.1. Suppose M is a smooth closed oriented 4-manifold which contains as a
connected summandeither:
(1) a K3 surface,
(2) or a product F × F of surfaces with at least one of the genera g or h strictly
g h
larger than17.
Then thegrouphomomorphismDiff(M) → π Diff(M) does notadmitasection.
0
In the process we prove a theorem (Theorem 1.2 below) about the cohomology of
BDiff(M) forM aK3 surface.
1.1. Strategy of the proof. For a closed oriented 4-manifold M, let Q denote the
M
symmetric bilinear form on H2(M;Z)/torsion coming from the cup product, and let
Aut(Q ) denote the group of automorphisms of H2(M;Z)/torsion which preserve
M
Q . Sendingadiffeomorphismtotheinducedautomorphismoncohomologyinducesa
M
grouphomomorphism
π Diff(M) → Aut(Q ).
0 M
The group Aut(Q ) is an arithmetic group, and hence its real cohomology is com-
M
putableinarangeofdegreesdependingontherankandsignatureofM (seesection3for
a precise statement). The pullbacks of the known classes on BAut(Q ) to BDiff(M)
M
can be identified with certain geometrically constructed classes, and the Bott Vanishing
Theorem [Bot70] implies that many of these geometric classes vanish on BDiff(M)δ
(where Diff(M)δ denotes the diffeomorphism group with the discrete topology). Since
anysection ofDiff(M) → π Diff(M) factors throughDiff(M)δ, nonexistenceofasec-
0
tionisimpliedbynontrivialityofan appropriatecohomologyclasson Bπ Diff(M).
0
The above argument is essentially due to Morita [Mor87]. The difficulty in applying it
is in proving that one of the obstruction classes is nonzero on the mapping class group.
When M is a surface there are a multitude of methods which prove the nontriviality of
theseobstructions,buttheyallsharearelianceonHarer’shomologicalstabilitytheorem
[Har85] for mapping class groups of surfaces. There is no known analogue of this the-
orem in dimension 4 or above. Thus, for 4-manifolds (other than products of surfaces)
onemustfind anewtechniqueto provethenontrivialityoftheobstructionclasses.
Our method is to detect the obstruction classes, when M is the 4-manifold underlying
a K3 surface, by pullingthem back to the “homotopy modulispace” M (M) of unit
Ein
volumeEinsteinmetricsonM. Whilethediffeomorphismgroupisratherintractable,the
topologyofthismodulispacehasamorerigidcharacterandinfactitcanbecompletely
understoodviatheGlobalTorelliTheorem of[Loo81].
LetEin(M)denotethespaceofunitvolumeEinsteinmetricsonM,withtheC∞ topol-
ogy;thereis acontinuousand properaction ofDiff(M) (thisaction isnotfree), and the
homotopy quotient M (M) := Ein(M) × EDiff(M) is the aforementioned
Ein Diff(M)
“homotopy moduli space”. Collapsing Ein(M) to a point gives a map M (M) →
Ein
BDiff(M). For a K3 surface, Borel’s work [Bor77] shows that H∗(BAut(Q );R) is
M
isomorphicto
H∗(BO ;R) ∼= R[p ] ∼= R[ch ,ch ]/(ch2 = 12ch )
3 1 4 8 4 8
THEDIFFEOMORPHISMGROUPOFAK3SURFACEANDNIELSENREALIZATION 3
in degrees ∗ ≤ 9 (where ch is the component of the Chern character in degree i). We
i
showinsection5,
Theorem 1.2. ForM a K3 surface,thecomposition
M (M) → BDiff(M) → Bπ Diff(M) → BAut(Q )
Ein 0 M
is injectiveonreal cohomologyindegrees ∗ ≤ 9.
Theclasscorrespondingthedegree8componentoftheCherncharacteronBO provides
3
an obstruction to the Nielsen realization problem when M is a K3 surface. A simple
argument given in section 2.2 extends the nontriviality of this obstruction from the K3
surface tomanifoldscontainingaK3 surface asa connectedsummand.
1.2. Relation to the stable mapping class group of 4-manifolds. We begin by re-
calling a part of the theory of surface mapping class groups. Given a surface F, one
can stabilize by letting the genus tend to infinity (i.e. repeatedly forming the connected
sum with a torus). The colimit of the resulting system of mapping class groups is the
stable mapping class group of F. By the solution to Mumford’s conjecture [MW07,
GMTW08], the rational cohomologyof the stable mapping class group is a polynomial
algebra on generators κ of degree2i. Harer-Ivanov stabilityis the fundamentaltool for
i
relating the stable mapping class group to unstable mapping class groups—it says that
the cohomology of π Diff(F) is isomorphic to the cohomology of the stable mapping
0
class groupina rangeofdegreesproportionaltothegenusofF.
In dimension4, the outlinesofan analogous story beginto emerge. Givena 4-manifold
M onecan stabilizebyrepeatedly formingtheconnected sum witha fixed manifoldX.
The stable mapping class group of M is the colimit of the resulting directed system of
2
mapping class groups. By [Gia08], when M is simplyconnected and X = CP2#CP ,
theresultingstablemappingclass groupis actuallyindependentofM andis infact iso-
morphic to the stabilization of the group Aut(Q ). The rational cohomology of this
M
stabilized group has been computed by Borel (see section 3) and it is is a polynomial
algebra on classes ℓ of degree 4i. The problem remains to decide how much of this
i
polynomial algebra one can see in the cohomology of the mapping class group of the
unstabilizedmanifoldM. In dimension4 no theorem analogous to Harer-Ivanov stabil-
ityexistsorisevenconjectured. However,Theorem 1.2(togetherwithProposition2.10
below)canbeintepretedasimplyingthatwhenM containsaK3surfaceasaconnected
summand then the first two generators of the polynomial algebra are nonzero (although
itcan happenthat thereisalinearrelationbetween ℓ and ℓ2).
2 1
Organization of the paper. In section 2 we define characteristic classes in the coho-
mology of BDiff(M) which provide potential obstructions to Nielsen realization, and
we show that these classes can bepulled back from BAut(Q ). We also providesome
M
bootstrappingtools for extendingnontrivialityresults. In section 3 we recall somefacts
about the automorphism groups of unimodular forms and the real cohomology of these
groups. In section 4 we recall some facts about Einstein metrics on a K3 surface and
apply the Global Torelli Theorem to understand M (M) and relate it to Aut(Q ).
Ein M
4 JEFFREYGIANSIRACUSA
In section 5 we study this relation on cohomology and prove Theorem 1.2, from which
Theorem 1.1follows.
Acknowledgments. This work was inspired by Morita’s beautiful paper [Mor87]. The
ideaofusingK3modulispacesgrewoutofaconversationwithAravindAsokandBrent
Doran, and much of my mathematical perspective is derived from them. Comments
from Eduard Looijenga, Andrew Dancer, and Ulrike Tillmann helped considerably as
this manuscriptdeveloped,and I thank Peter Kronheimerfor pointingout an error in an
earlierversion. IgratefullyacknowledgethesupportofanNSFgraduatefellowshipand
thehospitalityoftheIHES.
2. CHARACTERISTIC CLASSES AND THE NIELSEN REALIZATION PROBLEM
Let M bea smoothclosed oriented manifoldofdimension4k. In thissection wedefine
analogues ℓ ∈ H4i(BDiff(M);Q) oftheMiller-Morita-Mumfordcharacteristicclasses
i
for4k-dimensionalmanifolds. Weobserve,followingMorita’sapplication[Mor87,Sec-
tion 8] of the Bott Vanishing Theorem [Bot70], that these classes provide potential ob-
structionsto Nielsenrealization.
Definition 2.1. ForM a4k-dimensionalmanifold,definecharacteristicclasses
ℓ := π L (TνE) ∈ H4i(BDiff(M),Q),
i ∗ i+k
e
where π is the integration along the fibres map for the universal M-bundle E →
∗
BDiff(M). Wewriteℓ = ℓ .
Pi i
HereTνE istheverticaltangentbundleoftheuniversalM-bundle,and L isthedegree
i
4i component of the Atiyah-Singer modification of Hirzebruch’s L-classe. (The L class
correspondsto theformalpowerseries x .) e
tanh(x/2)
Wewillseeshortlythattheℓ classescan bethoughtofaslivingonthediffeomorphism
i
group, the mapping class group, or even the group of automorphisms of the middle
cohomology.
Let Q denotetheunimodularsymmetricbilinearform
M
Q : H2k(M;Z)/torsion⊗H2k(M;Z)/torsion → H4k(M;Z) ∼= Z
M
given by the cup product pairing. Let p = b+, q = b− be the dimensions of the maxi-
2k 2k
mal positiveand negativedefinite subspaces of H2k(M;R). We write Aut(Q ) for the
M
group of automorphismsof H2k(M;Z)/torsion which preserve Q . Sending a diffeo-
M
morphismtotheinducedautomorphismoncohomologydefinesagrouphomomorphism
Diff(M) → Aut(Q ).
M
Let ch ∈ H∗(BO ;R) be the total Chern character (it is a rational power series in the
p
Pontrjagin classes, or equivalently, it is the pullback of the Chern character on BU by
p
the map induced by sending a real vector bundle to its complexification), and let ch
i
denotethecomponentindegreei. Considerthecomposition
proj
(1) BDiff(M) → Bπ Diff(M) → BAut(Q ) → BO ≃ BO ×BO −→ BO .
0 M p,q p q p
THEDIFFEOMORPHISMGROUPOFAK3SURFACEANDNIELSENREALIZATION 5
Proposition2.2. In H∗(BDiff(M);R) thefollowingrelationholds:
ℓ = pullbackvia (1)of 2ch .
i 4i
Remark 2.3. We will therefore abuse notation and simply write ℓ for the pullback of
i
2ch to any of BAut(Q ), Bπ Diff(M), or BDiff(M). Note that when M is a K3
4i M 0
surface then p = 3 in the map (1), and on BO the relation ch2 = 12ch holds, and so
3 4 8
ℓ2 = 12ℓ .
1 2
Proof. This is a consequence of the Atiyah-Singer families index theorem. Consider a
fibre bundle M → E → B, and let η denote the associated vector bundle formed by
replacingM withH2k(M,R). AchoiceofafibrewiseRiemannianmetriconE induces
a Hodgestaroperator ∗ : H2k(M;R) → H2k(M;R) which satisfies ∗2 = 1. Hence this
bundle splits as a sum of positive and negative eigenspaces η = η ⊕η . The Atiyah-
+ −
Singerindextheoremforfamiliesapplied tothesignatureoperatorgivestheequation
ch(η −η ) = π L(TνE)
+ − ∗
e
(see[AS68,Section6]and[AS71,Theorem5.1]). Therealvectorbundleηhasstructure
group Aut(Q ), which is a discrete group. Hence η is flat, and so by the Chern-Weil
M
construction all Pontrjagin classes of η vanish (see e.g. [MS74, p. 308, Corollary 2]).
Therefore 0 = ch(η) = ch(η )+ch(η ) and so
+ −
(2) ℓ(E) = π L(TνE) = ch(η −η ) = 2ch(η ).
∗ + − +
e
Finally,observethatthecharacteristicclassesofthebundleη coincidewiththeclasses
+
pulledback from BO alongthecompositionof(1). (cid:3)
p
LetDiff(M)δ denotethediffeomorphismgroupendowedwiththediscretetopologyand
considerthenatural mapǫ : BDiff(M)δ → BDiff(M).
Theorem 2.4. FordimM = 4k andi > k therelation
ǫ∗ℓ = 0
i
holdsinH∗(BDiff(M)δ;R).
Proof. Morita’s argument [Mor87, Theorem 8.1] when dimM = 2 carries over verba-
timinthe4k dimensionalsetting;weincludeitforcompleteness. ThespaceBDiff(M)δ
is the classifying space for smooth M bundles which are flat, which is to say bundles
equippedwithafoliationtransversetothefibresandofcodimensionequaltothedimen-
sionofM (theprojectionofeachleafofthefoliationdowntothebaseisacoveringmap).
Let M → E → B be a fibre bundle with structure group Diff(M)δ and let F denote
the corresponding foliation. Then the normal bundle to F can be canonically identi-
fiedwiththeverticaltangentbundle. NowBott’sVanishingTheorem[Bot70]statesthat
the rational Pontrjagin ring of TνE vanishes in degrees greater than 8k. In particular,
L (TνE) = 0for4(i+k) > 8k,andthereforeℓ (E) = 0fori > k. Finally,sincethis
i+k i
heoldsforanyflatM-bundlewherethebaseandtotalspacearemanifolds,itholdsinthe
universalcase onBDiff(M)δ. (cid:3)
6 JEFFREYGIANSIRACUSA
Corollary2.5. ForM a 4k-dimensionalmanifold,theclasses
ℓ ∈ H4i(Bπ Diff(M);R)
i 0
(respectively, in H4i(BAut(Q );R)) for i > k are potential obstructions to the exis-
M
tence of a section of the grouphomomorphism Diff(M) → π Diff(M) (respespectively
0
Diff(M) → Aut(Q )). Thatis,iftheseclassesarenonzerothenasectioncannotexist.
M
Proof. Existence of such a section means that the identity on π Diff(M) (respectively
0
Aut(Q )) factors through Diff(M); since π Diff(M) (respectively Aut(Q )) is dis-
M 0 M
creteitactuallyfactorsthroughDiff(M)δ. ByProposition2.2theℓ classesonBDiff(M)
i
arepulledbackfromtheℓ classesonBπ Diff(M) orBAut(Q ),andbyTheorem2.4
i 0 M
theyarezerowhenpulledbacktoBDiff(M)δ. HenceiftheyarenonzeroonBπ Diff(M)
0
(respectivelyBAut(Q ))then nosectioncan exist. (cid:3)
M
Remark2.6. Hilsum[Hil89]providesaversionoftheIndexTheoremwhichisvalideven
in the Lipschitz setting, and the proof of Bott’s Vanishing Theorem works verbatim in
theC2 setting(althoughitisunknownifBott’stheoremholdsintheLipschitzcategory);
hence Corollary 2.5 still holds if Diff(M) is replaced by the C2 diffeomorphismgroup.
However, as Morita points out, the above method provides no information about lifting
mappingclass groupstohomeomorphismsinlightofthetheoremofThurston-McDuff-
Mather(seeforexample[McD80])thatBHomeo(M)δ → BHomeo(M) isahomology
isomorphism.
The remainder of this paper will be concerned with the question of nontriviality and
algebraicindependenceoftheℓ classes.
i
2.1. A product of surfaces. If M4k is a product of an even number of surfaces then
it is easy to derive nontriviality and algebraic independence of the ℓ classes from the
i
the known nontriviality and algebraic independence of the Miller-Morita-Mumford κ
i
classes forsurfaces.
SupposeM = F ×···×F , whereF isaclosed surfaceofgenusg .
g1 g2k gi i
Proposition 2.7. The ring homomorphism R[ℓ ,ℓ ,...] → H∗(BDiff(M);R) is injec-
1 2
tiveindegrees∗ ≤ max({g })/2−1. ThesameholdsforBDiff(MrD4k,∂(MrD4k)).
i
Proof. Letπ : E → BDiff(F )betheuniversalF -bundleandconsidertheℓ classes
i i gi gi j
oftheproductbundle
E = E −Q→πi BDiff(F ),
Y i Y gi
whichhasfibre F . Theverticaltangentbundlecanbewrittenasanexternalproduct
TνE ∼= TνEQ. Hegnice
i
Q
ℓ(E) = (π ×···×π ) L(TνE) =(π ×···×π ) L(TνE )×···×L(TνE )
1 2k ∗ 1 2k ∗h 1 2k i
e e e
=(π ) L(TνE )×···×(π ) L(TνE )
1 ∗ 1 2k ∗ 2k
e e
THEDIFFEOMORPHISMGROUPOFAK3SURFACEANDNIELSENREALIZATION 7
SinceTνE isarank 2 vectorbundle,L (TνE ) = (constant)·(e(TνE ))2j and so
i j i i
e
(π ) L (TνE ) = (constant)·κ .
i ∗ j i 2j−1
e
Thereforeℓ(E)isalinearcombinationofexternalproductsoftheκclassesofthesurface
bundlesE
i
Theκ classesarealgebraicallyindependentinH∗(BDiff(F );R)uptodegreeg/2−1;
i g
this is because the cohomology agrees with the stable (i.e g → ∞) cohomology in this
range byHarer-Ivanov stability[Iva93]and theκ classes areknownto bealgebraically
i
independent in the limit g → ∞ [Mum83, Mor87, Mil86]. Hence the classes ℓ (E) are
i
nontrivialand algebraicallyindependentup tothedesireddegree.
By Harer-Ivanov stability the κ classes remain algebraically independent up to degree
i
g/2−1 when pulled back to BDiff(F rD2,∂(F rD2)). Naturality ofthe ℓ classes
g g i
togetherwiththeinclusion
Diff(F rD2,∂) ֒→ Diff(M rD4k,∂)
Y gi
nowimpliesthesecond partoftheproposition. (cid:3)
Notethatwhenk = 1thefirstobstructionclassforNielsenrealizationonthe4-manifold
F × F is the class ℓ in degree 8. Hence Nielsen realization fails if one of g or h is
g h 2
strictlylargerthan 17.
2.2. Connected sums. Let M ,...M be4k-manifoldseach havinga(4k −1)-sphere
1 n
as boundary, and let Diff(M ,∂M ) denote the group of diffeomorphisms which fix a
i i
collar neighborhood of the boundary pointwise. By a slight abuse of notation, we write
M #···#M for the closed manifold created by gluing each M onto the boundary
1 n i
components of a 4k-sphere with the interiors of n discs deleted. Extending diffeomor-
phismsby theidentityonthepuncturedsphereinduces amap
µ : BDiff(M ,∂M )×···×BDiff(M ,∂M ) → BDiff(M #···#M ).
1 1 n n 1 n
Lemma 2.8. Theclassµ∗ℓ (M #···#M )isgivenbythesumoverj oftheproductof
i 1 n
ℓ on theBDiff(M ,∂M ) factorand1 on eachof theotherfactors;i.e.
i j j
j−1 n−j
n
µ∗ℓ (M #···#M ) = 1×···×1×ℓ ×1×···×1.
i 1 n Xz }| { i z }| {
j=1
In particular, if ℓ is nontrivial on BDiff(M ,∂M ) for some j then ℓ is nontrivial on
i j j i
BDiff(M #···#M ). Thesameholdswith Diff replaced byπ Diff.
1 n 0
Proof. Thisfollowsimmediatelyfromthecommutativesquare
µ
BDiff(M ,∂M )×···×BDiff(M ,∂M ) // BDiff(M #···#M )
1 1 n n 1 n
(cid:15)(cid:15) (cid:15)(cid:15)
BAut(Q )×···×BAut(Q ) // BAut(Q ⊕···⊕Q )
M1 Mn M1 Mn
togetherwithProposition2.2. (cid:3)
8 JEFFREYGIANSIRACUSA
Thisnext theoremshowsthat fora givenmanifoldM with boundaryasphere, nontrivi-
alityoftheℓ classesinarangeofdegreesonM impliesapartialalgebraicindependence
i
on iteratedconnected sumsM#···#M.
Theorem 2.9. Suppose ∂M ∼= S4k−1 and suppose that the classes ℓ ,...,ℓ are all
1 n
nonzero in H∗(BDiff(M,∂M);R). Then the monomials {ℓm1 ···ℓmn| m ≤ N} are
1 n i
P
all linearly independent in H∗(BDiff(#NM);R). In particular, on #NM the classes
ℓ ,...,ℓ satisfy no polynomial relations of degree ≤ 4kN. This holds also for Diff
1 n
replaced bythemappingclassgroupπ Diff.
0
Proof. Definethelengthofasimpletensor
a ⊗···⊗a ∈ H∗(BDiff(M,∂M;R)⊗···⊗H∗(BDiff(M,∂M;R)
1 N
∼= H∗(BDiff(M,∂M)×···×BDiff(M,∂M);R)
tobethenumberofcomponentsa whicharenotscalar(i.e. degree0). Lemma2.8yields
i
an expression of µ∗(ℓm1 ···ℓmn) as a sum of simple tensors. Since m ≤ N, one
1 n i
P
observes that the maximallength terms in this expressionare precisely all permutations
of
ℓ ⊗···⊗ℓ ⊗ℓ ⊗···⊗ℓ ⊗···⊗ℓ ⊗···ℓ ⊗1⊗···⊗1.
1 1 2 2 n n
| m{z1 } | m{z2 } | m{zn } | N−{Pzmi }
By considering these maximal length terms, we see that the classes µ∗(ℓm1 ···ℓmn) are
1 n
all linearlyindependent. (cid:3)
Suppposeoneknowsthatsomeoftheℓ arenontrivialforaclosedmanifoldM. Inorder
i
to apply Lemma 2.8 to show that these classes are nontrivial for a manifold containing
M asaconnectedsummandwemustshowthattheℓ classespullbacknontriviallyalong
i
themapBDiff(MrD,∂(MrD)) → BDiff(M). IfM isaproductofsurfacesthenthis
isalreadyaccomplishedinProposition2.7. Ontheotherhand,ifM issimplyconnected
then the kernel of π Diff(M r D,∂(M r D)) → π Diff(M) is either trivial or Z/2
0 0
since it is generated by the Dehn twist around the boundary sphere [Gia08, Proposition
3.1]. (Theterm’Dehntwist’herereferstotheimageofthenontrivialelementofπ SO
1 4k
underthemapSO → Diff(S4k−1).) Thus
4k
H∗(Bπ Diff(M rD,∂(M rD));Z[1/2]) ∼= H∗(Bπ Diff(M);Z[1/2]).
0 0
Proposition 2.10. If ℓ is nonzero on the mapping class group of a simply connected
i
closed 4k-dimensional manifold M, then ℓ is nonzero on the mapping class group of
i
anymanifoldcontainingM asa connected summand.
3. THE REAL COHOMOLOGY OF ARITHMETIC GROUPS
In this section we recall a technique due to Matsushima [Mat62] and Borel [Bor74,
Bor81] used to study the real cohomology of arithmetic groups such as automorphism
groups of unimodularlattices. Then we relate the classes produced by this techniqueto
theℓ classesstudiedin theprevioussection.
i
THEDIFFEOMORPHISMGROUPOFAK3SURFACEANDNIELSENREALIZATION 9
3.1. The Borel-Matsushima homomorphism. First we review the general construc-
tion, due to Borel and Matsushima, of a homomorphism from the cohomology of a
compact symmetric space to the cohomology of a related arithmetic group. The real
cohomologyof thecompact symmetricspace is easily computableand we shall refer to
theclassesin theimageofthishomomorphismastheBorel-Matsushimaclasses.
Suppose G is a connected semisimplelinear Lie group and A ⊂ G a discrete subgroup
for which we would like to understand the cohomology with real coefficients. We have
in mind G = G(R) for an algebraic group G and A = G(Z) the arithmetic subgroup
givenby theintegerpointsin G
The group G admits a maximal compact subgroup K; let X = G/K be the associated
symmetric space of non-compact type. The discrete group A acts on X from the left
with finite isotropy subgroups and X is contractible, so H∗(A\X;R) ∼= H∗(BA;R).
Let Gu be a maximal compact subgroup of the complexification GC which contains K.
The quotientX = G /K is a compact symmetricspaceknown as thecompact dual of
u u
X. Matsushima[Mat62]defined aring homomorphism
(3) j∗ : H∗(X ;R) → H∗(A\X;R) ∼= H∗(BA;R)
u
and studied the extent to which this map is injective and/or surjective when A is a co-
compact subgroup. Borel [Bor74] later extended these results to the case of general
arithmeticsubgroups. We refer toj∗ as theBorel-Matsushimahomomorphism.
We now briefly review the the construction of the Borel-Matsushima homomorphism;
this will be needed in the proof of Lemma 3.3 below. The cohomology of A\X can be
computed using de Rham cohomology. If A is torsion free then the de Rham complex
Ω∗(A\X) is easily seen to be isomorphic as a dga to the ring Ω(X)A of A-invariant
forms on X, and when A is not torsion free it is true (by a standard argument) that
Ω∗(X)A stillcomputesthecohomologyofA\X.
AneasywaytoproduceA-invariantformsonX istotakeG-invariantformsonX. The
inclusion
(4) Ω∗(X)G ֒→ Ω∗(X)A
induces a map on cohomology. A G-invariant form on X is entirely determined by
its value on the tangent space at a single point since G acts transitively, and hence the
complex Ω∗(X)G is entirely a Lie algebra theoretic object. Let g, g , k denote the Lie
u
algebras ofG, G ,K respectively. Then thereareCartan decompositions
u
g ∼=k⊕p
g ∼=k⊕ip,
u
and hencethereare canonicalisomorphisms
K
(5) Ω∗(X)G ∼= p∗ ∼= Ω∗(X )Gu.
(cid:16)^ (cid:17) u
Since X is a compact manifoldand Ω∗(X )Gu consistsofharmonicforms, Hodgethe-
u u
ory impliesthat
(6) Ω∗(X )Gu ∼= H∗(X ;R).
u u
10 JEFFREYGIANSIRACUSA
Combining(4), (5), and (6), oneobtainsthehomomorphism(3).
Borel proved that this homomorphism is injective and surjective in ranges of degrees
depending only on the root system of G. In particular, for the B and D root systems
n n
wehave:
Theorem 3.1([Bor81, Theorem 4.4]). ForAan arithmeticsubgroupofa groupG with
rootsystemoftypeD (resp. B ),j∗ isbijectiveindegrees∗ < n−1(resp. ∗ < n)and
n n
injectivefor∗ = n−1(resp. ∗ = n).
Remark 3.2. The group SO+ has root system of type D if p + q is even, and
p,q (p+q)/2
B ifp+q isodd. ThereforethebijectiverangeforSO+ is∗ ≤ ⌊(p+q)/2⌋−2.
⌊(p+q)/2⌋ p,q
3.2. AreinterpretationofBorel-Matsushima. Wenowgiveaninterpretation(Lemma
3.3) of the Borel-Matsushima classes on the level of maps between classifying spaces.
Proposition3.6willfollowfrom thistogetherwithBorel’s Theorem 3.1above.
Precomposition of j∗ with the classifying map c : X → BK for the principal K-
u u
bundleG → G /K = X givesa homomorphism
u u u
H∗(BK;R) → H∗(BA;R).
On theotherhand, onehas
A ֒→ G ≃ K
which alsoinducesamap fromthecohomologyofBK to thecohomologyofBA.
Lemma 3.3. These two homomorphismscoincide.
We will need the following result for the proof of this proposition. The principal K-
bundle G → G /K = X is classified by a map c : X → BK. Suppose A is
u u u u u
torsion free, so A\G → A\G/K = A\X is a principal K bundle classified by a map
c : A\X → BK. In thissituationwehave:
Lemma 3.4([Bor77], Proposition7.2). Thediagram
c∗
H∗(BK;R) u // H∗(X ;R)
u
c∗ (cid:15)(cid:15) wwoooooojo∗ooo
H∗(A\X;R)
commutes.
ProofofLemma 3.3. By a well-known result of Selberg, the arithmetic group A admits
afiniteindexsubgroupAwhichistorsionfree. Since
e
H∗(BA;R) ∼= H∗(BA;R)A/Ae ⊂ H∗(BA;R),
e e
and the Borel-Matsushima homomorphism is natural with respect to inclusions, it suf-
fices to verify the claim for torsion free arithmeticgroups. So we now assume that A is
torsionfree.