Milnor-Wood inequalities for products

0.15 MB·English

by  Michelle Bucher

MILNOR-WOOD INEQUALITIES FOR PRODUCTS
MICHELLE BUCHER AND TSACHIKGELANDER
2
1
Abstract. WeproveMilnor-woodinequalitiesforlocalprod-
0
2 ucts of manifolds. As a consequence, we establish the gener-
alizedChernConjectureforproductsM×Σk foranyproduct
n ofamanifoldM withaproductofk copiesofasurfaceΣfor
a
k sufficiently large.
J

1

]
T 1. Introduction
A
Let M be an n-dimensional topological manifold. Consider the
.
h Eulerclassε (ξ) ∈ Hn(M,R)andEulernumberχ(ξ) = hε (ξ),[M]i
n n
t
a of oriented Rn-vector bundles over M. We say that the manifold
m
M satisfies a Milnor-Wood inequality with constant c if for every
[ flat oriented Rn-vector bundles ξ over M, the inequality

1 |χ(ξ)| ≤ c·|χ(M)|
v
2 holds. Recallthatabundleisflatifitisinducedbyarepresentation
3 of the fundamental group π (M). We denote by MW(M) ∈ R∪
3 1
{+∞} the smallest such constant.
0
. If X is a simply connected Riemannian manifold, we denote by
1
0 M]W(X) ∈ R ∪ {+∞} the supremum of the values of MW(M)
2 when M runs over all closed quotients of X.
1
Milnor’s seminal inequality [Mi58] amounts to showing that the
:
v ]
Milnor-Woodconstant ofthehyperbolicplane HisMW(H) = 1/2,
i
X and in [BuGe11], we showed that M]W(Hn)= 1/2n.
r WeproveaproductformulafortheMilnor-Woodinequalityvalid
a
for any closed manifolds:
Theorem 1. For any pair of compact manifolds M ,M
1 2
MW(M ×M ) = MW(M )·MW(M ).
1 2 1 2
For the product formula for universal Milnor-Wood constant, we
restrict to Hadamard manifolds:
Theorem 2. Let X ,X be Hadamard manifolds. Then
1 2
MW(X ×X ) = MW(X )MW(X ).
1 2 1 2
Michelle Bucher acknowledges support from the Swiss National Science
g g g
Foundationgrant PP00P2-128309/1. Tseachik Gelanderacknowledgessupport
of the Israel Science Foundation and the European Research Council.
1
MILNOR-WOOD INEQUALITIES FOR PRODUCTS 2
OneimportantapplicationofMilnor-Woodinequalitiesistomake
progress on the generalized Chern Conjecture.
Conjecture 3 (Generalized Chern Conjecture). Let M be a closed
oriented aspherical manifold. If the tangent bundle TM of M ad-
mits a flat structure then χ(M) = 0.
Asthenameindicates, thisconjectureimpliestheclassicalChern
conjecture foraffine manifolds predicting the vanishing of the Euler
characteristicofaffinemanifolds. Thisisbecauseanaffinestructure
on M induces a flat structure on the tangent bundle TM.
As pointed out in [Mi58], if MW(M) < 1 then the Generalized
Chern Conjecture holds for M. Indeed, if χ(M) 6= 0 the inequality
|χ(M)| = |χ(TM)| ≤ MW(M)·|χ(M)| < |χ(M)|
leads to a contradiction.
One can use Theorem 1 to extend the family of manifolds satis-
fying the Generalized Chern Conjecture. For instance:
Corollary 4. Let M be a manifold with MW(M) < +∞. Then
the product M ×Σk, where Σ is a surface of genus ≥ 2 and k >
log (MW(X)) satisfies the Generalized Chern Conjecture. In par-
2
ticular, if χ(M) 6= 0, then M ×Σk does not admit an affine struc-
ture.
Remark 5. 1. One can replace Σk in Corollary 4 by any Hk-
manifold.
2. The corollary is somehow dual to a question of Yves Benoist
[Be00, Section 3, p. 19] asking wether for every closed manifold
M there exists m such that M × Sm admits an affine structure.
For example, for any hyperbolic manifold M, the product M ×S1
admits an affine structure, but in general m = 1 is not enough.
Indeed, Sp(2,1) has nonontrivial9-dimensionalrepresentations and
the dimension of the associated symmetric space is 8.
Note that since there are only finitely many isomorphism classes
of oriented Rn-bundles which admit aflatstructure, itisimmediate
that the set
{|χ(ξ)| |ξ is a flat oriented Rn−bundle over M}
isfiniteforevery M. Inparticular, ifχ(M) 6= 0, thereexistsafinite
Milnor-Wood constant MW(M) < +∞.
However, in general, the Milnor-Wood constant can be infinite.
Indeed, the implication χ(M) = 0⇒ χ(ξ)= 0, for ξ a flat oriented
Rn-bundle, does not hold in general. In Section 5 we exhibit a flat
bundle ξ with χ(ξ) 6= 0 over a manifold M with χ(M) = 0. This
example is inspired by Smillie’s counterexample of the Generalized
MILNOR-WOOD INEQUALITIES FOR PRODUCTS 3
Chern Conjecture [Sm77] for nonaspherical manifolds, and likewise
this manifold is nonaspherical.
The following questions are quite natural:
(1) Does there exist a finite constant c(n) depending on n only
such that MW(M) ≤ c(n) for every closed aspherical n-
manifold?
(2) Let X be a contractible Riemannian manifold such that
there exists a closed X-manifold M with MW(M) < ∞. Is
]
MW(X) necessarily finite?
(3) Does χ(M) = 0 ⇒ χ(ξ) = 0 for flat t oriented Rn-bundles
ξ over aspherical manifolds M?
2. Representations of products
Lemma 6. Let H ,H be groups and ρ : H × H → GL (R) a
1 2 1 2 n
representation of the direct product and suppose that ρ(H ) is non-
i
amenable for both i = 1,2. Then, up to replacing the H ’s by finite
i
index subgroups, either
• V = Rn decomposes as an invariant direct sum V = V′⊕V′′
where the restriction ρ|V′ = ρ′ ⊗ρ′ is a nontrivial tensor
1 2
representation, or
• V = V ⊕V where G is scalar on V .
1 2 i i
Proof. Thiscanbeeasilydeducedfromtheproofof[BuGe11,Propo-
sition 6.1]. (cid:3)
Proposition 7. Let H = k H be a direct product of groups and
i=1 i
let ρ : H → GL+(R) be an orientable representation, where n =
n
Q
k
m . Suppose that ρ(H ) is nonamenable for every i. Then, up
i=1 i i
to replacing the H ’s by finite index subgroups H′ = k H′, either
P i i=1 i
(1) there exists 1 ≤ i < k such that V = Rn decomposes non-
0 Q
trivially to an invariant direct sum V = V′ ⊕ V′′ and the
restricted representation ρ|(H′ ×Q H′,V′)
i0 i>i0 i
H′ × H′ −→ GL(V′)
i0 i
iY>i0
is a nontrivial tensor, or
(2) the representation ρ′ factors through
+
k k
ρ′ : Hi′ −→ GLm′(R) −→ GL+n(R),
i !
i=1 i=1
Y Y
where the latter homomorphism is, up to conjugation, the
canonicaldiagonalembedding, andρ′(H′)restrictstoascalar
i
representation on each GL (R), for i 6= j.
mj
MILNOR-WOOD INEQUALITIES FOR PRODUCTS 4
Moreover, if all m are even then either m′ < m for some i or one
i i i
can replace GL with GL+ everywhere.
+
The notation ki=1GLm′i(R) stands for the intersection of
ki=1GLm′(R) w(cid:16)itQh the positive (cid:17)determinant matrices.
i
QProof. We argue by induction on k. For k = 2 the alternative is
immediate from Lemma 6. Suppose k > 2. If Item (1) does not
hold,itfollowsfromLemma6that,uptoreplacingtheH ’sbysome
i
finite index subgroups, V decomposes invariantly to V = V ⊕V′
1 1
where ρ(H ) is scalar on V′ and ρ( H ) is scalar on V . We
1 1 i>1 i 1
now apply the induction hypothesis for H restricted to V′.
Finally, in Case (2), since m =Qn, eii>t1herim′ < m for so1me
i i i
Q
i or equality holds everywhere. In the later case, if all the m ’s
i
P
are even, given g ∈ H , since the restriction of ρ(g) each V is
i j6=i
scalar, it has positive determinant. We deduce that also ρ(g)| has
Vi
positive determinant. (cid:3)
3. Multiplicativity of the Milnor-Wood constant for
product manifolds – A proof of Theorem 1
Let M ,M be two arbitrary manifolds. We prove that
1 2
MW(M ×M ) = MW(M )·MW(M ).
1 2 1 2
FirstnotethattheinequalityMW(M ×M )≥ MW(M )·MW(M )
1 2 1 2
is trivial. Indeed, let ξ ,ξ beflat oriented bundles over M andM
1 2 1 2
respectively of the right dimension such that |χ(ξ )| = MW(M )·
i i
|χ(M )| for i = 1,2. Then ξ ×ξ is a flat bundle over M ×M
i 1 2 1 2
with
|χ(ξ ×ξ )| = |χ(ξ )||χ(ξ )| = MW(M )·MW(M )·|χ(M ×M )|.
1 2 1 2 1 2 1 2
For the other inequality, let ξ be a flat oriented Rn-bundle over
M ×M , where n = Dim(M )+Dim(M ). We need to show that
1 2 1 2
|χ(ξ)| ≤ MW(X )·MW(X )·|χ(M)|.
1 2
Observe that if we replace M by a finite cover, and the bundle ξ by
its pullback to the cover, then both sides of the previous inequality
are multiplied by the degree of the covering.
The flat bundle ξ is induced by a representation
ρ: π (M ×M ) ∼=π (M )×π (M ) −→ GL+(R).
1 1 2 1 1 1 2 n
If ρ(π (M )) is amenable for i = 1 or 2, then ρ∗(ε ) = 0 [BuGe11,
1 i n
Lemma4.3]andhenceχ(ξ) = 0andthereisnothingtoprove. Thus,
we can without loss of generality suppose that, upon replacing Γ by
afiniteindexsubgrouptherepresentationρfactorsasinProposition
7.
MILNOR-WOOD INEQUALITIES FOR PRODUCTS 5
In case (1) of the proposition, we obtain that ρ∗(ε ) = 0 by
n
Lemma 10 and [BuGe11, Lemma 4.2]. In case (2) we get that ρ
factors through
ρ: π1(M1)×π1(M2)−→ GLm′1(R)×GLm′2(R) + −→i GL+n(R),
where the latter embeddin(cid:16)g i is up to conjugati(cid:17)on the canonical
embedding. Furthermore, up to replacing ρ by a representation in
the same connected component of
+
Rep(π1(M1)×π1(M2), GLm′1(R)×GLm′2(R) )
which will have no influence on(cid:16) the pullback of the(cid:17)Euler class,
we can without loss of generality suppose that the scalar repre-
sentations of π1(M1) on GLm′2 and π1(M2) on GLm′1 are trivial,
so that ρ is a product representation. If m′ or m′ is odd, then
1 2
i∗(εn) = 0 ∈Hcn((GLm′1(R)×GLm′2(R))+). If m′1 and m′2 are both
even then Proposition 7 further tells us that either m′ < m for
i i
i = 1 or 2, or the image of ρ lies in GL+ (R)×GL+ (R). In the
m1 m2
first case, the Euler class vanishes [BuGe11, Lemma 4.2], while in
thesecond case, we immediately obtain thedesired inequality. This
finishes the proof of Theorem 1.
4. Multiplicativity of the universal Milnor-Wood
constant for Hadamard manifolds - a proof of
Theorem 2
Theorem 2 can be reformulated as follows:
Theorem 8. Let X be a Hadamard manifold with de-Rham decom-
k k
position X = X , then MW(X) = MW(X ).
i=1 i i=1 i
We shall noQw prove Theorem 8. NotQe that the inequality "≥"
g g
is obvious. Let M = Γ\X be a compact X-manifold. We must
k
show that MW(M) ≤ MW(X ). Note that Γ is torsion free.
i=1 i
Let us also assume that k ≥ 2. If M is reducible one can argue
Q
by induction using Theoremg1. Thus we may assume that M is
irreducible. Observe that this implies that Isom(X) is not discrete.
If Γ admits a nontrivial normal abelian subgroup then by the flat
torus theorem (see [BH99, Ch. 7]) X admits an Euclidian factor
whichimpliesthevanishingoftheEulerclass. Assumingthatthisis
not thecase we apply theFarb–Weinberger theorem [FaWe08, The-
orem 1.3] to deduce that X is a symmetric space of non-compact
type. Thus, up to replacing M by a finite cover (equivalently, re-
place Γ by a finite index subgroup), we may assume that Γ lies
in G = Isom(X)◦ = k Isom(X )◦ = k G and G is an ad-
i=1 i i=1 i
joint semisimple Lie group without compact factors and Γ ≤ G is
Q Q
irreducible in the sense that its projection to each factor is dense.
MILNOR-WOOD INEQUALITIES FOR PRODUCTS 6
k
Denote by G the universal cover of G , and by Γ ≤ G the
i i i=1 i
pullback of Γ.
Let ρ : Γe→ GL+(R) be a representation indeucingQa flaet ori-
n
ented vector bundle ξ over M. Up to replacing Γ by a finite in-
dex subgroup, we may suppose that ρ(Γ) is Zariski connected. Let
S ≤ GL+(R) be the semisimple part of the Zariski closure of ρ(Γ),
n
and let ρ′ :Γ → S be the quotient representation. By superrigidity,
the map Ad◦ρ′ :Γ → Ad(S) extends to φ :Γ ≤ k G → Ad(S)
i=1 i
(see[Ma91,Mo06,GKM08]). This map canbepulled to φ: Γ → S.
Q
k k
Recall also that G is a central discrete extension of G
i=1 i i=1 i
and, likewise, Γ is a central extension of Γ. If n = deimXe and
Q i Qi
n = k n we deduece from Proposition 7 and Lemma 10 that
i=1 i
either the Euleer class vanishes or the image of φ lies (up to decom-
P
posing the vector space Rn properly) in ( k GL )+.
i=1 ni
Suppose that MW(X ) is finite for all i = 1,.e..,k and let M be
i i
closed X -manifolds. Let ξ′ be the flat vQector bundle on k M
i i=1 i
coming from ρ reduced to k M , and let ξ′ be the vector bundle
i=1 i i Q
on M induced by ρ , i= 1,...,k. By Lemma 9, we have
i i
Q
χ(ξ) e χ(ξ′) k χ(ξ′) k
= e = i ≤ MW(X ),
vol(M) vol( ki=1Mi) i=1 vol(Mi) i=1 i
Y Y
which finishes the proQof of Theorem 8. (cid:3)
5. Example: a flat bundle with nonzero Euler number
over a manifold with zero Euler characteristic
Recall that given two closed manifolds of even dimension, the
Euler characteristic of connected sums behaves as
χ(M ♯M ) = χ(M )+χ(M )−2.
1 2 1 2
The idea is to find M = M ♯M such that M admits a flat bundle
1 2 1
with nontrivial Euler number in turn inducing such abundle on the
connected sum, and tochoosethen M insuch away that theEuler
2
characteristic of the connected sum vanishes. Take thus
M = Σ ×Σ , M = (S1×S3)♯(S1×S3)and M = M ♯M .
1 2 2 2 1 2
These manifolds have the following Euler characteristics:
χ(M ) = 4,
1
χ(M ) = 2χ(S1 ×S3)−2 = −2,
2
χ(M) = 0.
Let η be a flat bundle over Σ with Euler number χ(η) = 1. (Note
2
thatweknowthatsuchabundleexistsby[Mi58].) Letf : M → M
1
MILNOR-WOOD INEQUALITIES FOR PRODUCTS 7
be a degree 1 map obtained by sending M to a point, and consider
2
ξ = f∗(η×η).
Obviously, since η is flat, so is the product η ×η and its pullback
by f. Moreover, the Euler number of ξ is
χ(ξ) = χ(η×η) = 1.
Indeed, the Euler number of η×η is the index of a generic section
ofthe bundle, which we canchoosetobenonzero on f(M ), sothat
2
we can pull it back to a generic section of ξ which will clearly have
the same index as the initial section on η×η.
6. Proportionality principles and vanishing of the
Euler class of tensor products
Lemma 9. Let X be a simply connected Riemannian manifold,
G = Isom(M) and ρ: G → GL+(R) a representation. Then χ(ξρ) ,
n vol(M)
where M = Γ\X is a closed X-manifold and ξ is the flat vector
ρ
bundle induced on M by ρ restricted to Γ, is a constant independent
of M.
Proof. There is a canonical isomorphism H∗(G) ∼= H∗(Ω∗(X)G))
c
betweenthecontinuous cohomologyofGandthecohomologyofthe
cocomplex of G-invariant differential forms Ω∗(X)G on X equipped
with its standard differential. (For G a semisimple Lie group, every
G-invariant form is closed, hence one further has H∗(Ω∗(X)G) ∼=
Ω∗(X)G.) In particular, in top dimension n = dim(X), the coho-
mology groups are 1-dimensional Hn(G) ∼= Hn(Ω∗(X)G)) ∼= R and
c
contain the cohomology class given by the volume form ω .
X
Since the bundle ξ over M is induced by ρ, its Euler class ε (ξ )
ρ n ρ
is the image of ε ∈ Hn(GL+(R,n) under
n c
Hn(GL+(R,n)−−ρ→∗ Hn(G) −→ Hn(Γ) ∼= Hn(M),
c c
where the middle map is induced by the inclusion Γ ֒→ G. In
particular, ρ∗(ε ) = λ·[ω ] ∈ Hn(G) for some λ ∈ R independent
n X c
of M. It follows that χ(ξ )/Vol(M) = λ. (cid:3)
ρ
Lemma 10. Let ρ : GL+(n,R) × GL+(m,R) → GL+(nm,R)
⊗
denote the tensor representation. If n,m ≥ 2, then
ρ∗(ε ) = 0∈ Hnm(GL(n,R)×GL(m,R)).
⊗ nm c
Proof. The case n = m = 2 was proven in [BuGe11, Lemma 4.1],
basedonthesimpleobservationthatinterchangingthetwoGL+(2,R)
factors does not change the sign of the top dimensional cohomology
class in H4(GL(2,R)×GL(2,R)) ∼= R, but it changes the orienta-
c
tion on the tensor product, and hence the sign of the Euler class in
H4(GL+(4,R)).
c
MILNOR-WOOD INEQUALITIES FOR PRODUCTS 8
Let us now suppose that at least one of n,m is strictly greater
than 2, or equivalently, that n+m < nm. The Euler class is in the
image of the natural map
Hnm(BGL(nm,R)) −→ Hnm(GL(nm,R)).
c
By naturality, we have a commutative diagram
Hnm(BGL+(nm,R)) // Hnm(GL+(nm,R))
c
ρ∗ ρ∗
⊗ ⊗
(cid:15)(cid:15) (cid:15)(cid:15)
Hnm(B(GL+(n,R)×GL+(m,R))) // Hnm(GL+(n,R)×GL+(m,R))).
c
Since the image of the lower horizontal arrow is contained in degree
≤ n+m, it follows that ρ∗(ε ) =0. (cid:3)
⊗ nm
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