Michael W. Davis* ...... covered by Euclidean space, to appear in Ann. of Math. 117(1983). ... B. Mazur, A note on some contractible 4-manifolds, Ann. of Math. (2).
COXETER GROUPS AND ASPHERICALMANIFOLDS Michael W. Davis*
I.
Contractible manifolds and aspherical manifolds. Suppose that
Mn
is a compact, contractible n-manifold with boundary.
These assumptions imply that the boundary of
M
has the homology of an
(n-l)-sphere; however, they do not imply that it is simply connected. n ~ 3, that
then for ~M
Mn
to be homeomorphic to a disk it is obviously necessary
be simply connected.
(cf. [5], [13], [18], [19]). examples of such In fact, if G
Mn
n ~ 6,
satisfying
If
For
n ~ 5
this condition is also sufficient
On the other hand, for
n ~ 4,
there exist
with non-simply cormected boundary (cf. Ill], [12], [14]). then the fundamental group of the boundary can be any group
HI(G) = 0 = H2(G)
A non-compact space
W
(cf. [9]).
is simply connected at
~
if every neighborhood of
(i.e., every complement of a compact set) contains a simply connected neighborhood of
~.
Suppose
W
is a locally compact, second countable,
Hausdorff space with one end (i.e., it is connected at written as an increasing union of compact sets C...,
and where each
W - Ci
is connected.
~).
W =~=I The space
Then
Ci
W
where
W
can be CICC2
~
is semi-stable if
the inverse sequence
~i (W-C1) + ~i (W-C2) + "'"
satisfies the Mittag-Leffler Condition, i.e., if there exists a subsequence of epimorphisms.
(This condition is independent of the choice of
semi-stable, then the isomorphism class of the inverse limit llm ~I(W-Ci) W
Ci.)
~
if and only if it is semi-stable and
trivial (cf. [6], [7], [17]). *Partially supported by NSF grant MCS-8108814(AOI).
W
is
~T (W) =
is independent of all choices (including base points).
is simply connected at
If
The space ~T (W)
is
198
Now suppose that for
Wn
Wn
is an open contractible n-manifold.
to be homeomorphic
connected at If
Wn
~.
For
to
n ~ 4
En
it is obviously necessary
this condition
and
~I (W) ~ w I (~M).
Hence,
M n,
in view of our previous
n = 3,
contractible
there is a well-known
example of Whitehead
Aspherical manifolds In such contexts
if its universal arise naturally
of its universal
exp : T M ÷ M x
(at any point
cover is diffeomorphic
with maximal compact subgroup subgroup,
then the universal
diffeomorphic believed
to Euclidean
is aspherlcal
space.
the following well-known
homeomorphic
to Euclidean space.
Of course,
curvature
T M ~ I n, x
and if
F C G
and
Mn
is any
then the exponential
3) if
F\G/K
(cf. [7],
(they exist), but rather the existence
G
hence,
is
[I0].
map
the
is any Lie group
G/K
discrete which is
some people
[8; p. 423]).
is
of exotic contractible
of exotic contractible
admit a group of covering transformations
Some positive results
e.g., in [6], [7],
is either
is any torsion-free
the issue here is not the existence
compact quotient.
2) if
On the basis of such examples conjecture
As examples
> 0
cover of any closed aspherical manifold
manifolds which simultaneously
contexts.
usually consists of a
is a covering projection;
cover of the manifold
The universal
obtained,
to
K
CONJECTURE.
manifolds
~.)
cover with Euclidean space.
sectional
x ~ M)
n ~ 4.
cover is contractible.
the plane or the interior of the disk, more generally,
universal
for any
i) the universal cover of a Riemann surface of genus
complete manifold of non-positive
is
remarks,
in a variety of geometric
the proof that the manifold
direct identification
Wn
[22] of an open
3-manifold which is not simply connected at
A space is aspherlcal
we have:
~,
then
(cf. [5],[20]).
then, clearly,
there exist examples which are not simply connected at (For
n ~ 3,
that it be simply
is also sufficient
is the interior of a compact manifold
semi-stable
If
(i.e., non-existence
with
results) have been
199
This paper, which is an expanded version of my lecture, is basically an exposition of some of the results of [3].
We shall discuss a method of [3] of
using the theory of Coxeter groups to construct a large number of new examples of closed aspherical manifolds.
Although the construction is quite classical,
its full potential had not been realized previously.
The most striking
consequence of the construction is the existence of counterexamples to the above conjecture in each dimension
~ 4.
In Section 2 of this paper we give some background material on Coxeter groups.
In Section 3 we explain the construction in dimension two, where it
reduces to the classical theory of groups generated by reflections on simply connected complete Riemann surfaces of constant curvature.
The main results are
explained in Section 4 where we consider the same construction in higher dimensions.
In Section 5 we discuss a modification of the construction which
gives many further examples. result
This modification is used in Section 6 to prove a
(the only new result in this paper) concerning the Novikov Conjecture.
Finally,
in Section 7 we discuss a conjecture concerning Euler characteristics
of even-dlmensional closed aspherical manifolds. 2.
Coxeter groups. In this section we review
some
standard material on Coxeter groups.
For
the complete details, see [2]. Let
~
be a finite graph (i.e., a l-dimenslonal finite simplicial
complex), with vertex set
V
and edge set
which assigns to each edge an integer
~ 2.
1 m(v,w) =
E
;
m({v,w}); ;
and let
m : E + Z
For each pair
if
v = w
if
{v,w} E E
otherwise.
These data give a presentation of a group:
r = ~.
reflection groups is as follows.
(cf.
choice of
of finite Coxeter groups in dimension 3 can then be
interior angles is a spherical
groups
~
we
F.
The classification
REMARK 2.
H2
There is then a well-defined
This map is easily seen to be a covering projection.
recovered
at this
or the hyperbolic plane
through the sides of this convex polygon.
obtain a map H 2.
F
R 2,
can be realized as a convex polygon
with interior angles as specified by the labels. homomorphism
~/m
on
is
the sum of this interior angles is greater than,
equal to, or less than S 2,
2m
m
In the general hyperbolic
case,
F
206
In dimension
3 there is a rich theory and complete result due to Andreev
[I] or [21]).
In dimensions
general understanding
> 3
there are a few isolated examples but no
of the possibilities;
(something
like dimensions
hyperbolic
reflection
while in very high dimensions
> 30) Vinberg has apparently proved that cocompact
groups do not exist.
groups have representations
In summary,
as cocompact geometric
these cases, at least in dimensions
> 3,
The construction Let
X
a PL-triangulatlon two purposes.
The slmpllclal
(*)
L.
Second,
be the vertex set of
For each simplex
S • L
This means that for any
example,
if
holds,
~X
L,
E
will be given the structure of the
the edge set, and
~
the 1-skeleton.
m : E + {2,3,...}
to
is the following:
the subgroup
S ~ L
L
rS,
generated by
S,
if we discard the edges labelled
is finite.
2,
is an octahedron we could label its edges as below. complex
S
L
if we label every edge by
since in this case for each
The second condition
(*~) If
will be used for
then the
labelled graph is the Coxeter diagram of a finite Coxeter group.
for any simplicial (*)
L
be
L.
The first condition
resulting
complex
L
First, a Coxeter system will be constructed by labelling the
There are two conditions which we want our labelling satisfy.
then the
n.
of its boundary.
dual cell complex to V
As we shall
contractible n-manlfold with boundary and let
edges of the 1-skeleton of
Let
chambers.
simplicity.
in dimension
be a compact,
few Coxeter
there are very few combinatorial
if we drop our geometric requirements,
situation reverts to its original 4.
relatively
reflection groups and in
types of convex polyhedra which can occur as fundamental see in the next section,
(cf.
is a subset of
is the converse
V
such that
S G L
2,
we will have
FS
is finite,
then
In general,
then condition r S = (X/2X) S.
to the first:
S • L.
For
207
I
=
i v L
A\
is an octahedron with a vertex at
Since by construction an unordered pair of vertices and only if less than
m(v,w) < ~, 3.
S
(**)
means that if
triangle, p
-i
+q
-I
FS
is infinite.
then the labels +r
-i
L
belongs to
E
if
~
contains a suhgraph
which is isomorphic to the 1-skeleton of a simplex and which
is not equal to the 1-skeleton of a simplex in such that
{v,w}
this condition is vacuous for subsets of cardinality
Hence, condition
with vertex set
~.
For example, p,q,r
if
L, L
then the edge labels must be
is the suspension of a
on the edges of the triangle must satisfy
< i.
is the suspension of a triangle with a vertex at
m.
208
If any subgraph of
~
which is isomorphic
equal to the l-skeleton holds vacuously.
complex, [3]). L
3
Therefore,
conditions
by its barycentric
L,
then condition
(**)
the octahedron has this property as does any
edges.
then its barycentric
More generally,
if
L
is any simplicial
subdivision has this property (*)
and
(**)
(*)).
(cf. Lemma 11.3 in
are always satisfied
subdivision and label each edge
fashion which satisfies L
of some simplex in
For example,
polygon with more than
to the l-skeleton of a simplex is
2
if we replace
(or in any other
We now assume that we have labelled
in some fashion so that conditions
(*)
and
(**)
the edges of
hold and we let
(r,V)
denote the resulting Coxeter system. Next we cellulate
~X
as the dual cell complex.
is an octahedron,
X
dual cell of
and for each simplex
S.
{v}
will be a cube.
For each S ~ L,
Thus, for example,
v • V, let
XS
let
X
v
if
denote the
be the dual cell of
Thus,
X S = v~ES Xv
(The
Xv,
X.)
which are faces of codimension
Also,
for each subset
S
of
V
one in
X,
are called the panels of
put
Xo(s) = v~,SXv.
If S
(D)
S
is actually a simplex of
in the barycentric
If
S E L,
zero in
For each x
and let
xE
subdivision
L, of
then L.
then the union of panels
X (S)
is a regular neighborhood
of
Hence,
X (S)
is a disk of codimension
3X.
X
FV(x)
let
V(x) = {v ~ Vlx ~ X } v
be the subgroup generated
be the set of panels which contain V(x).
As before, we define a
L
209
F-space
~=
(F×X)/-~,
as in the previous
where the equivalence
section.
The map
which we regard as an inclusion. for
F
on
~
and that
b)
relation
x + [l,x]
induces an embedding
Observe that
for each
a)
x e X
--J is defined exactly
X
X + 2~.
is a fundamental
the isotropy subgroup
domain
is
FV(x).
We claim that: (I)
F
acts properly on
(2)
~
is a manifold and
(3)
~
is contractible.
(4)
If
L
(=ax)
F
acts locally smoothly.
is not simply connected,
connected at
(I) is equivalent
condition
(**).
to condition
(*),
X
intersect
we can find a neighborhood
where
gm
is a neighborhood
of
R V(x) •
U x
Let
x in
simplicial
cone in
of
x
~.
if
(i) the orbit space is Hausdorff,
FV(x) on
7~/r m X,
x
of
is
R m x l v(x).
(i) holds. [g,x]
C v(x).
~(x)
(3)
intersect
is a simplicial
Hence,
X
of the form
and where
complex.
proving and
C v(x)
is the standard neighborhood if and only
under the isotropy
(*)
(iii).
implies Hence,
of itself•
(ii).
r
Also,
acts properly.
gW x
is a
Since
chamber for its canonical action
m RV(x)
and
Wx = rv(x)Ux
is locally Euclidean and that the action is
(2).
(4).
respect to the generating through the panels of
Hence,
Rm × cV(x)
be the corresponding
which is invariant
rv(x)CV(X) ~
in
This means that for each
from all other translates
satisfying
to
(ii) each isotropy group is finite and
Condition
This shows that
locally linear, Proof of
(3) is equivalent
in
is a finite Coxeter group, a fundamental
Z V(x)
is not simply
Recall that an action of a discrete group is proper
subgroup and which is disjoint
neighborhood
of
W x = rv(x)Ux
(iii) each point has a neighborhood
Since
while
in general position.
x e X
in
?~
To say that the cells of a polyhedron
general position means that the dual polyhedron the panels of
then
=.
Basically,
Proof of (i) and (2).
2~.
gX
For any
g • F
set
Let
V.
(i.e.,
let Vg
£(g)
denote its word length with
denote the set of "reflections"
V g = gvg-l.)
Put
210
C(g) = {w ~ vgl£(wg) < £(g)}
and
B(g) = (v ~ V I £(gv) < £(g)} = g-iC(g)g.
(C(g)
is the set of reflections across panels of
image of
gX
is closer to
X
than is
gX.
B(g)
gX
such that the reflected
is the set of reflections
across these same panels after they have been translated back to
X.)
Also, put
~(gX) = gX (B(g)),
i.e.,
6(gX)
is the union of those panels of
gX
LEMMA A (cf. [3, Lemma 7.12] or [16, p. 108]). PB(g )
which are indexed by
For any
g ~ F,
C(g).
th___£esubgroup
is finite.
Sketch of Proof.
Finite Coxeter groups are distinguished from inf~nlte Coxeter
groups by the fact that each finite one has a unique element of longest length. It is not hard to see that
rB(g )
has such an element.
the (unique) element of shortest length in the coset from Exercises 3 and 22, p. 43, in [2] that length in
rB(g ).
a = gh -I
Explicitly, let
gFB(g ).
h
be
Then it follows
is the element of longest Q.E.D.
211
Next order the elements of
F,
gl,g2, . . . .
so that
~(gi+l ) ~ A(gi).
Since
1
gl = I.
For each integer
m ~ I,
is the unique element of length
O,
put
X m = gm X'
6Xm = 6(gmX),
m
= ~Jx Tm i= I
i.
The next lemma asserts that the chambers intersect as one would expect them to. (For a proof see [3, Lemma 8.2].)
LEMMA B.
Thus,
For each integer
Tm
m ~ 2, X m O
is obtained from
Tm_ 1
Tm_ 1 = 6Xm.
by pasting on a copy of
X
along a certain
union of panels. By Lemma A, for each implies that Xo(B(g))
B(g) ~ L.
m
the group
this means that
6Xm
Tm.
is finite.
Condition
Since
~
~X
for each
g &r.
Since
is a disk of codimension zero in
is the boundary connected sum of
contractible, so is
rB(g )
(**)
Statement (D) then implies that the union of panels
is a disk of codimension zero in
gmXo(B(gm ),
T
g • P
m
copies of
= Um=l
T m,
X. ~
Since
X
6x m =
8Xm.
Hence
is
is contractible, which
proves 3). Since
~
contractible) to Since
~T
m
dim X ~ 3)
is formed by successively pasting on copies of T
m
along disks,
is the connected sum of
that
~l(~Tm)
Z~m
T
m
X
(which are
is homotopy equivalent to
copies of
i s t h e f r e e p r o d u c t of
~X,
m
~T . m
we have (provided
c o p i e s of
~I(~X).
212
Moreover,
the map
~l(~-Tm+l)
~ ~l(~Tm+l)
inclusion can clearly be identified factors of the free product. Thus,
~
is semi-stable
÷ ~l(?~!-Tm)
with the projection
In particular,
generated).
and the inverse limit
REMARK I.
connected whenever Section
Let
pair
2,
(H,V)
{v,w}
(Z/2Z) V. V.
U/F'
dim X ~ 4,
statement
F'
is finite,
F'
Thus,
Namely,
Thus,
H
leads to a closed ~X
may be non-simply
of a torsion-free
S
REMARK 3.
The construction
concerning
fundamental
group on
r
V
for each
such that
~IF s
S,
is contained
is the commutator description
(g,x)~'
= 2
which is the identity on
to a subgroup of some F'
subgroup
is the finite Coxeter group
for any such
conjugate
where
m(v,w)
is any subset of
hence,
(Incidentally,
~ (H×X)/N'
acts a reflection
that if the universal
H
~ : F + H
then there is an alternative ~/F'
torsion-free
is obtained by labelling each edge
Since any finite subgroup of
is torsion-free.
= (F×X)/~ , ~/F'.
V.
If
F s m ((Z/2Z) S = HS; H S.
~I(~X)
(4) implies that the conjecture of
(F,V)
in
~.
isotropy group and is consequently that
F'
be the Coxeter system defined by setting
be the kernel of
isomorphism onto
if
(or even finitely
always contains
there is any easy construction
of distinct vertices
then
Hence,
~ 4.
There is a natural epimorphism
Let
F
In view of the fact that
In the special case where
of a graph by r'.
remarked,
1 is false in dimensions
REMARK 2.
m ~ i.
is the "projective
~I (~X).
of finite index and any such subgroup
aspherical manifold
m
(4).
As we have previously
subgroups
onto the first
~i (?'~)
then this inverse limit is not trivial
This proves
induced by
this map is onto for each
free product" of an infinite number of copies of is not trivial,
~ ~I(~T m)
(h,x) ~
2~/F' with quotient
FS
is an in some FS,
we have
subgroup.)
If
of the quotient g-lh eHv(x). X.
of this section suggests several questions
groups at
=.
First of all, R. Geoghegan has conjectured
cover of a finite complex has one end, then it must be
213
semi-stable.
After seeing the above construction, H. Sah and R. Schultz both
asked if the fundamental group at
=
of the universal cover of a closed
aspherical manifold can ever be non-trivial and finitely generated (i.e., can the universal cover be stable at
REMARK 4.
without being simply connected at
~).
There is some flexibility in the main construction of this section.
First of all, for X
~
be a cell.
~
to be contractible it is not necessary that each face of
All that the proof requires is that
each proper face be acyclic.
be contractible and that
(For example, we can change a panel of
taking connected sun with a homology sphere.)
All that is required is that
L
to be a L
then be "dualized"
(i.e., "resolved")
with contractible faces. as before.
This
X
by
PL
triangulation of
have the homology of an
(n-1)-sphere and that it be a polyhedral homology manifold each k-simplex must have the homology of an
X
Secondly and more interestingly,
it is not necessary for the simplicial complex a homology sphere.
X
(i.e., the link of
(n-k-2)-sphere).
Any such
L
to produce a contractible n-manifold
can X
can then be used to produce a contractible
If one is interested in constructing proper, locally smooth
actions of a discrete group generated by "reflections" on a contractible manifold with compact quotient,
then there is no further flexibility.
That is
to say, with the above two provisos, every cocompact reflection group on a contractible manifold can be constructed as above (cf. [3]). shall see in the next section,
However, as we
there is quite a bit more flexibility if we are
only interested in producing more examples of aspherical manifolds. 5.
~ seneralization. We begin this section by considering a modification of our construction in
dimension 2.
Rather than starting with the underlying space of
let it be any compact surface with nonempty boundary.
a 2-disk,
Take a polygonal sub-
division of the boundary and label the vertices by integers X
X
~ 2.
For example,
could be one of the "orbifolds" pictured below with random labels on the
vertices.
The polygonal subdivision of the boundary together with the labelling
214
defines a Coxeter system obtain a surface
Z~
(F,V).
As before, paste together copies of
(=(F×X)/~)
longer be contractible,
with
F-action.
but it will be aspherical
(After all, almost every surface is aspherical.) torsion-free aspherical
subgroup of finite index in
F,
The surface provided If
then
X
be a compact aspherical n-manifold
replacing
L
Let
L
be a triangulation
by its barycentric
subdivision,
in an arbitrary dimension.
of
BX.
(F,V)
sum of copies of
(the wedge of two aspherlcal ~
torsion-free
~( X.
~
Since
X
its fundamental
F,
then
so is
The fundamental
Wl(X).
~/F'
to the
is aspherical,
spaces is again aspherical).
subgroup of finite index in
(*) and
with proper F-action.
is homeomorphic
is an infinite free product of copies of
aspherical manifold;
After possibly
satisfies conditions
As in Section 4, there results an n-manifold
infinite boundary connected
(The boundary need
we can find a labelling of its
The proof of Claim (3) in Section 4 shows that
by
is not a 2-disk.
will be a closed
with boundary.
edges so that the resulting Coxeter system
group of
will no
surface.
not be aspherical.)
(**).
to
is any closed
U/F'
Next let us try to make the same modification Let
T'
X
Z~
X
If
F'
is a
is a closed
group is, of course, an extension of
r'
~i ( ~ ) .
REMARK.
One can imagine situations where it would be convenient
if one could
double a compact aspherical manifold along its boundary and obtain a closed aspherical manifold as the result.
However,
the doubled manifold
is not
215
aspherical unless the boundary into the fundamental
is aspherical
and its fundamental
group of the original manifold.
can be viewed as a fancy method of doubling
group injects
The method described above
so that the result will be
aspherieal. In [21] Thurston considers aspherical 3-manifolds. of Sections 4 and 5.) 3-dimensional
(In fact, the discussion
technical
for certain compact
in [21] inspired the results
Thurston shows that with a few more hypotheses
orbifold can be given a hyperbolic
"double" certain hyperbolic sub-surfaces
the above construction
3-manifolds
of their boundaries).
structure.
This allows him to
along their boundaries
This "orbifold
the
(or actually
trick" plays an important
role in the proof of his famous theorem on atoroidal Haken
3-manifolds. 6.
An observation concerning the Novikov Conjecture. A group is geometrically
finite if it is the fundamental
aspherical
finite complex
aspherical
compact manifold with boundary).
If
Mn
(or equivalently,
if it is the fundamental
is a manifold with boundary with fundamental
is a surgery map homomorphism;
~ : [(M,~M);(G/TOP,*)]
however,
÷ Ln(Z).
it is if we replace
group of an
M
by
group
The map
s
7,
group of an
then there
need not be a
M × D i, i > O,
and
~M
by
B(M×Di). The Novikov Conjecture Mn
is any aspherical
[MxI,B(Mxl),(G/TOP,*)] the rationals.
Conjecture)
finite group
compact manifold with fundamental + Ln+l(Z )
becomes a monomorphism
A stronger version of this asserts that
(See [4] for a discussion
PROPOSITION.
for a geometrically
~
group
asserts that if ~,
then
o :
after tensoring with o
is a monomorphism.
of these conjectures.)
If the Novikov Conjecture holds for the fundamental
then it holds for every geometrically
(resp. the strong version of the Novikov
group of every closed aspherical manifold, finite group.
*The fact that the construction of the previous section could be used to prove this proposition first came up during a conversation with John Morgan.
216
Proof. ~.
Let
(X,~X)
Triangulate
2
be an aspherical
~X,
take the barycentric
to form a Coxeter system
Z~ = (FxX)/~,
and
compact manifold with fundamental
(F,V).
gg' = ~ / F '
There is a commutative
subdivision,
Let
H = (Z/2%) V,
= (H×X)/-w',
( G / T O P , * ) ] O'-~ L (~) n + i,
[~/,.', G/TOP]
point, where
÷ X/~X
(If by
o
and where
× D4
CLAIM.
and
The map
Proof of Claim. argument [3]).
L (~') n
by
outside
X
to a
is the map induced by the inclusion
follows easily from the next claim.
then replace
[Z~',G/TOP]
A
i,
= ~'.The proposition
is not a homomorphism,
~'
o
is the map which collapses everything
~' = Zl('~'),
= ~I(X)~-+ ~I(Z~')
F' = ker(F+H),
be as in Remark 2 of Section 4.
+ A*
A : U'
and label each edge by
diagram [(X,~X),
where
group
X
by
X × D 4, ~X
by
~(X×D4),
~'
[ ( ~ ' x D 4, Z4.'×$3); (G/TOP,*)].)
: [(X,~X),(G/TOP,*)]
+ [~',GITOP]
We first proved the corresponding
is a monomorphism.
statement
in cohomology°
The
is based on the existence of an "alternation map" (cf. Section 9 in
If
~
is a singular chain in
X,
then we can "alternate"
it to form the
chain
A(cO =
in
~'.
The map
chain map
~ ÷ A(=)
A : C,(X,~X)
~ h•H
clearly vanishes
÷ C ,(Z4J ).
,
which is a splitting for
split monomorphism space
Y,
on cohomology
[Y,G/TOP] ~ ~ m ~ H
rationally a monomorphism
on
4*
on
A
*
C,(~X);
hence,
Hence,
(with arbitrary coefficients).
G/TOP,
: H (~')
÷
A
is a
Since for any
this is enough to prove that
[(X,~X),(G/TOP,*)].
of the homotopy type of
A
,
: H (X,~X) + H (Z~').
(Y;~),
there is a
*
for the weak version of the Novikov Conjecture.) calculation
• c~
This induces a homomorphism
*
H (X,~X)
( - 1 ) £ ' h ' h~(
A
*
is
(Hence, the proposition holds According
showing that
to Sullivan's A
is a monomorphism
217
is equivalent to showing that it is a monomorphism on KO (
) ~ Z[½].
We have a l r e a d y p r o v e d i t
arbitrary coefficients.
H (
,Z(2))
and on
f o r o r d i n a r y cohomology w i t h
To prove the corresponding statement for an
extraordinary cohomology theory we first need to make some small modifications. Let
R
be a ring in which
invertible).
IHI
2
is
Put
~ = IHI -I
For any
is invertible (i.e., in which
R[H]-module
M
Z (-l)£(h)h ~ R[H]. hEH
define a submodule
MAlt = {z~Mlvz = -z,Vv~V}.
It is
easily checked that (a)
~2 =
(b)
MAlt = Image(~:M+M).
For an arbitrary cohomology theory it is not clear how to split However, suppose that 2
)
is invertible in
the singular set.
(c)
~*(
~*(~',Z)
is a cohomology theory with value in R
(e.g.
~*(
) = KO (
m
Z ~ h~H
(X,~X)
Z4
be
(~',Z),
we find that
Hence, there is a map
: ~*(7~')
7.
Z C
2@ * (x, ~x).
7~ * ( ~ , ,Z) Alt.
A .
Let
from which it follows that
Using (a), (b), and the sequence of the pair
which splits
R-modules and
By excision,
(d) ~ * ( Z g ' , Z ) Alt
~*(ZL,)AIt
) ~ Z[½]).
A .
+ ~ * ( Z ~ ' ) Alt = ~
(X,~X)
This proves the claim and consequently, the proposition.
The rational Euler characteristic and some conjectures. Associated to any orbifold there is a rational number, called its "Euler
characteristic," which is multiplicative with respect to orbifold coverings (cf. [21]).
If the orbifold is cellulated so that each stratum is a subcomplex, then
this is defined as the alternating sum of the number of cells in each dimension
218
where each cell is given a weight of the inverse of the order of its isotropy group. We suppose, L
as usual,
is a triangulation
of
labelling the edges of that Let
?.~: (FxX)/~. e(X)
that ~X,
L,
Xn
is a compact n-manifold with boundary,
that
(F,V)
is a Coxeter system obtained by
that condition
For each
S • L,
(*)
let
of Section 4 is satisfied,
X S ~ ~X
be the ordinary Euler characteristic
space of
X
minus that of
8X.
that
be the dual cell of
of the underlying
and
S.
topological
The rational Euler characteristic
of
X
is
then defined by
dim X S (I)
E
x(X) : e(X) +
S£L = e(X) + (-i) n
where
IFSI
-~E (-I) Card(S) 1 S6L T~T
denotes the order of
torsion-free Also, if
1
(-i)
It is then clear that if
FS •
subgroup of finite index in
?J- is contractible,
Euler characteristic For example,
if
then
F,
then
x(X) = X(F),
x(ZUF') where
=
X(F)
F'
is a
[r:r']x(X). is the rational
as defined in [16]. X
is a triangle and
~
is the
(p,q,r)-triangle
group,
then
3
1
1
1
1
-I
x(X) = i - ~ + (T~p + 7~q + ~7r) = 7 ((p
If
M2
is a closed surface which is aspherical,
follows that if surfaces,
+q-1+r-t)-1)"
then
N 2k = M 2I x ... x M k 2
then
x(M 2) ~ 0.
It
is a product of closed aspherical
(-I)kx(M 2k) > 0.
A well-known
conjecture of Hopf is the following:
CONJECTURE
1 (Hopf).
curvature,
then
If
M 2k
is a closed manifold of non-positive
(-I)kx(M 2k) > 0.
sectional
219
This was proved by Chern for 4-manifolds and by Serre [16] for local symmetric spaces.
Recently, H. G. Donnelly and F. Xavier have established it under a
hypothesis of pinched negative curvature. More generally W. Thurston has asked if the following conjecture is true.
CONJECTURE 2 (Thurston).
If
M 2k
is a closed aspherical manifold, then
(-1)k×(M 2k) > O.
It should be pointed out that this conjecture contradicts another conjecture of Kan-Thurston which asserts that any closed manifold has the same homology as some closed aspherical manifold. Conjecture 2 implies the following conjecture.
CONJECTURE 3.
Le___~tx 2 k , ~ , ~
Euler characteristic.
If
be as above and let
×(X)
~J. i__ssaspherical, then
denote the rational
(-l)k×(x 2k) ~ 0.
One might try to construct a 4-dimensional counterexample to the above conjecture as follows. of (**)
S 3.
Let
X
be a 4-cell and
Label the edges in
L
in some fashion so that conditions
hold and calculate
×(X)
using (i).
L
some specific triangulation (*)
and
After making a number of such
calculations and having the result invariably come out non-negative, I now believe Conjectures 2 and 3 are true.
A more or less random example of such a
calculation is included below.
EXAMPLE. J. (*).
Let
J
be a triangulation of
Label each edge in
J
by
2
S2
and let
and each edge in
L
be the suspension of
L-J
by
3.
This satisfies
If
J
is not the boundary of a tetrahedron and if each circuit of length
three in
J
bounds a triangle then it also satisfies
the number of i-simplices in
(i)
x(X) = 1 +
J.
Let us calculate
E (-I)Card(S)IPSI-I S~L
(**).
×(X)
Let
using
ai
denote
220
There are
(a0+2)
1 - ~ (ao+2).
contribute )
and
vertices
2a 0
in
L
= ~ =
There are
of type
i (weight ~);
~
(weight i__) 24 ; 2a 2
tetrahedra
a2
hence,
i ~
each of type
a 2.
aI
of type
1 (weight ~)
°'"
a0 - a I + a2 = 2
and
"
and
i (weight i-~);
~
(weight
1 1 ~ a I + ~ a 0. 2a I
of
i a2+ Ti~ a3). - (~
contribute
hence,
the
Thus,
1 1 1 1 1 x(X) = i - ~(a0+2 ) + (5 al+ ~ a0) - (8 a2+ ~ a l ) 1 ii = ~ ( - a 0 + a l - ]-~a2).
Since
the vertices
the edges contribute
of type
hence the triangles
tetrahedron
contribute
hence,
There are two types of edges:
There are two types of triangles: type
i ~;
each of weight
3a 2 = 2a 1,
we can rewrite
1 + ~
a2
this as
1 (5ai_2). x(X) = ~ 24
If this is to be a I ~ 9, or
< 0,
then the equation
(5,9).
1 a0 - ~ a I = 2
X(x)
> 1
In either case
J
if
J
that either
a I ~ 9.
is the suspension
If
(a0,a 1) = (4,6)
is the boundary
of a tetra-
of the boundary
does not hold; while
(**)
Thus,
of a
in every other case
O.
The geometric cellulated
picture of
X
as the dual polyhedron
Combinatorially, and
implies
The first case can only happen
hedron and the second only if triangle.
48 a I ! - - ~ < i0.
then we must have
y x {i}
X
is
y x I;
meet each (face of
is as follows. to
J
Let
Y
be a 3-cell w i t h
and with all dihedral angles
however, Y) ~ I
the top and bottom faces at a dihedral
angle of
~Y
90 ° . y x {0}
60 ° .
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E . M . Andreev, On convex polyhedra Sbornik 10(1970) No. 5, 413-440.
[2]
N. Bourbaki, 1968.
Groupes e t Alsebres
in Lobacevskii
de Lie, Chapters
spaces, Math. U S S R
IV-VI, Hermann,
Paris
221
[3]
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[4]
F. T. Farrell and W. C. Hsiang, On Novikov's conjecture for non-positively curved manifolds, I. Ann. of Math. 113(1981), 199-209.
[5]
M. Freedman, The topology of four-dimensional manifolds, J. Diff. Geom. 17(1982), 357-453.
[6]
B. Jackson, End invariants of group extensions, Topology 21(1981), 71-81.
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F. E. A. Johnson, Manifolds of homotopy type Phil. Soc. 75(1974), 165-173.
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M. A. Kervaire, Smooth homology spheres and their fundamental groups. Amer. Math. Soc. Trans. 144(1969), 67-72.
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B. Mazur, A note on some contractible 4-manifolds, Ann. of Math. 73(1961), 221-228.
[12]
M. H. A. Newman, Boundaries of ULC sets in Euclidean n-space. N.A.S. 34(1948), 193-196.
[13]
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II, Proc. Cambridge
(2)
Proc.
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[14]
V. Po~naru, Les decompositions de l'hypercube en produit topologique, Bull. Soc. Math. France 88(1960), 113-129.
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A. Selberg, On discontinuous groups in higher dimensional symmetric spaces, Int. Colloquim on Function Theory, Tata Institute, Bombay, 1960.
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J. P. Serre, Cohomologie des groupes discrets, Prospects in Mathematics, Ann. Math. Studies Vol. 70 (pp. 77-169), Princeton University Press, Princeton 1971.
[17]
L. C. Siebenmann, The obstruction to finding a boundary for an open manifold of dimension ~ 5, thesis, Princeton 1965.
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S. Smale, Generalized Poincare's conjecture in dimensions greater than four, Ann. of Math. (2) 74(1961), 391-406.
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J. Stallings, Polyhedral homotopy-spheres, 66(1960), 485-488.
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Bull. Amer. Math. Soc.
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[22]
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