Topological finiteness theorems for manifolds
in Gromov-Hausdorff space
Steven C. Ferry
SUNY at Binghamton
April 26, 1993
Abstract. We give general conditions under which precompact sets of topolo*
*gical mani-
folds in Gromov-Hausdorff space contain finitely many homeomorphism types.*
* The main
result says that this is true if the manifolds in the set have a common co*
*ntractibility
function. We also show that there are at most finitely many closed n-manif*
*olds, n 6= 3,
with a common cell-like image.
1. Introduction
In [GPW1], Grove, Petersen, and Wu used comparison theory and controlled topo*
*logy to
prove that for any given n 6= 3 there are at most finitely many homeomorphism t*
*ypes of
Riemannian n-manifolds with any fixed lower bound on sectional curvature, lower*
* bound
on volume, and upper bound on diameter. One of the key steps in their argument *
*is the
theorem that all of the manifolds in such a class have a common "contractibilit*
*y function."
Here is the definition.
Definition 1.1. A function, ae : [0; R] ! [0; 1) which is continuous at 0 with *
*ae(0) = 0,
and ae(t) t for all t is a contractibility function for a metric space X if fo*
*r each x 2 X,
and t R, the metric ball of radius t centered at x contracts to a point in the*
* concentric
ball of radius ae(t).
The purpose of this paper is to extend their work by investigating conditions*
* under which
collections of manifolds with a common contractibility function contain only fi*
*nitely many
homemorphism types. The main theorem proves that this is true when n 6= 3 and *
*the
collection has compact closure in Gromov-Hausdorff space.
Definition 1.2.
(i)If X and Y are compact subsets of a metric space Z, the Hausdorff distanc*
*e between
X and Y is
dHZ(X; Y ) = inf{ffl > 0 | X Nffl(Y ); Y Nffl(X)}:
(ii)If X and Y are compact metric spaces, the Gromov-Hausdorff distance from*
* X to Y
is
dGH (X; Y ) = infZ{dHZ(X; Y )}
_________________________
Partially supported by NSF Grant DMS 9003746. The author would also like to tha*
*nk the University
of Chicago and the Institute for Advanced Study for support while some of this *
*work was accomplished.
Typeset by LAMS-*
*TEX
1
2
where X and Y are isometrically embedded in some Z.
(iii)Let CM be the set of isometry classes of compact metric spaces with the *
*Gromov-
Hausdorff metric.
It is well-known that CM is a complete metric space. See [G] or [P2] for an e*
*xposition.
It is important to note that the number R is part of the data of the contract*
*ibility
function.
Definition 1.3. Let Mman (n; ae) be the set of all (X; d) 2 CM such that X is *
*a topological
n-manifold with (topological) metric d so that ae is a contractibility function*
* for X.
Our main result is a finiteness theorem in Mman (n; ae).
Theorem 1. If n 6= 3 and C is a subset of Mman (n; ae) such that the closure o*
*f C is compact
in CM, then C contains only finitely many homeomorphism types of topological ma*
*nifolds.
It is well-known [HM, KS] that in dimensions 5 there are at most finitely ma*
*ny smooth
manifolds homeomorphic to a given topological manifold, so we have:
Corollary 1.4. If n 5 and C is a subset of Mman (n; ae) whose elements are sm*
*ooth
manifolds and the closure of C is compact in CM, then C contains only finitely *
*many
diffeomorphism types.
Remark 1.5.
(i)The n 6= 3 condition is necessary in the sense that if there is a counter*
*example to
the 3-dimensional Poincare Conjecture, then Theorem 1 is false in dimensi*
*on 3. See
Example 4.2.
(ii)Theorem 1 appears as "Main Theorem" on p. 206 of [GPW1]. Unfortunately, *
*the
proof given there contains an error, which was discovered in [M]. See als*
*o [GPW2].
The error goes back to the proof of the Theorem on p. 393 of [P1], where *
*it is as-
serted that a compact infinite-dimensional metric space must have finite-*
*dimensional
subsets of arbitrarily large dimension. This is not true, and Moore used *
*an example
of Dranishnikov [D] to show that a sequence of S7's in Mman (7; ae) can c*
*onverge to
an infinite-dimensional limit.
(iii)Such behavior cannot occur in the presence of a lower bound on sectional*
* curvature,
so the validity of Theorem A of [GPW1] is not affected by this error. See*
* [GPW2]
for details.
A second finiteness theorem follows from our Theorem 1 and work of Greene-Pet*
*ersen
[GrP].
Theorem 2. If n 4 and C(ae; V0; n) is the subset of Mman (n; ae) consisting o*
*f Riemannian
manifolds such that
V ol(M) V0 for every M 2 C(ae; V0; n);
3
then C(ae; V0; n) contains finitely many homeomorphism types. If n 5, then C *
*contains
only finitely many diffeomorphism types.
Our last main result proves "finiteness of resolutions" for "Dranishnikov man*
*ifolds!" Is it
possible that there is a resolution/obstruction-to-resolution theorem for, say,*
* locally simply
connected homology n-manifolds with finite cohomological dimension?
Theorem 3. For any topological space X, there are at most finitely many nonhome*
*omor-
CE
phic closed n-manifolds Mi, n 6= 3, so that Mi--! X.
CE CE
Remark 1.6. The proof shows that if M1 --! X and M2 --! X, then there is a homo*
*topy
equivalence f : M1 ! M1 preserving rational Pontrjagin classes. The main result*
* of [F2]
shows that f is a simple homotopy equivalence. If such maps exist with M1 and *
*M2
nonhomeomorphic, then it will show that there is a contractibility function ae *
*so that there
is a space X in CM so that there are copies of M1 and M2 with contractibility f*
*unction ae
which are arbitrarily close to X.
2. The proof of Theorem 1
The argument is by contradiction, so we assume that we have a sequence {Mi} 2
Mman (n; ae) so that all Mi's have distinct homeomorphism types. By the preco*
*mpact-
ness assumption, we can assume limMi= X in CM. Properties of the space X will p*
*lay a
major role in the argument. In general, X is a "Dranishnikov space," a space wi*
*th infinite
covering dimension and finite cohomological dimension. See [M] for examples.
Some useful properties of X
Definition 2.1. A map f : P ! Q between topological spaces is said to be (k + *
*1)-
connected if f# : ss`P ! ss`Q is an isomorphism for ` k and an epimorphism for*
* ` = k + 1.
The map f is said to be a weak homotopy equivalence if f is k-connected for all*
* k.
Here are the properties of X which will be used in the proof of Theorem 1.
` `
P1. There is a metric on Z = ( Mi) X which restricts to the given metrics *
*on X and
the Mi so that Z is compact and so that limMi= X in the Hausdorff metric *
*on Z.
4
P2. There is a neighborhood U of X in Z and a retraction r : U ! X. For i lar*
*ge, we
have Mi U and r|Mi: Mi! X is a weak homotopy equivalence.
P3. For each ffl > 0 there is an Nffl2 Z so that for each i; j Nfflthere are*
* homotopy
equivalences fij: Mi ! Mj with homotopy inverses gji: Mj ! Mi and homotop*
*ies
kijt: gjiO fij' id, hijt: fijO gji' id so that the maps fij, gjimove poin*
*ts less than
ffl and the tracks of the homotopies have diameters less than ffl in the *
*metric of Z.
P4. There are finite polyhedra P1 and P2 and maps
p2 ff
X -! P2 -! P1:
so that ff O p2 is (n + 3)-connected and so that p2 is (dim P1 + 3)-conne*
*cted.
To help the reader orient himself/herself, we will point out two properties w*
*hich X does
not have.
(i)There are no maps X ! Mi which are close to the identity in the metric of*
* Z. Such
maps would force X to be finite-dimensional.
(ii)There are no maps si : Pi ! X so that ff O p2 O s1 and p2 O s2 are homot*
*opic to the
identity.
Remark 2.2. The question of the existence of compact metric spaces with finite *
*coho-
mological dimension and infinite covering dimension was open for many years. E*
*dwards
[Walsh] proved that every such space is the cell-like image of a finite-dimensi*
*onal compact
metric space and Dranishnikov [D] exhibited the spaces themselves.
For now, we will skip the proof of properties P1-P4 and proceed with the proo*
*f of Theorem
1. The reader who can't wait should see x3.
Surgery-theoretic preliminaries
Definition 2.3. If M is a closed (topological) n-manifold, a homotopy structure*
* on M
is a closed n-manifold N together with a homotopy equivalence f : N ! M. Struc*
*tures
(N; f) and (N0; f0) are equivalent if there is a homeomorphism OE : N ! N0 so t*
*hat f0O OE is
homotopic to f.
N [ [ [f
| []
OE|~=
| aeoM
|uaeaeae0
N0 f
We denote the set of equivalence classes of homotopy structures on M by S(M).*
* In words,
a homotopy structure on M is a homotopy equivalence from a manifold N to M. Two*
* of
these are equivalent if there is a homeomorphism from one to the other making t*
*he diagram
homotopy commute. A structure "rel @" on (M; @M) is a homotopy equivalence (N; *
*@N) !
5
(M; @M) which is a homeomorphism on the boundary. Two of these are equivalent i*
*f there
is a homeomorphism OE : (N; @N) ! (N0; @N0) strictly commuting on the boundary *
*making
the diagram homotopy commute.
For n 5, S(M) can often be calculated using the Sullivan-Wall surgery exact *
*sequence
[W, Chapter 3 x1]:
. .!.Hn+1(M; F=T OP ) ! Ln+1(Zss1(M)) !
S(M) ! Hn(M; F=T OP ) ! Ln(Zss1(M)):
where "F=T OP " stands for the connective spectrum whose 0thspace is the infini*
*te loopspace
F=T OP . The groups Li(Zss1(M)) are the Wall surgery obstruction groups [Wa]. T*
*hey are
4-periodic and depend only on the fundamental group of M.
If we extend the notion of "homotopy structure" somewhat from the above, this*
* exact
sequence is covariantly functorial - even for polyhedra which are not necessari*
*ly manifolds.
We define a j-dimensional structure on a finite polyhedron P to be a (j + 4k)-d*
*imensional
structure rel boundary on a regular neighborhood of P in some (j + 4k)-dimensio*
*nal mani-
fold. That this is independent of k and the choice of manifold containing P is *
*a consequence
of Siebenmann periodicity. The procedure is reminiscent of the use of Bott peri*
*odicity to
extend the theory of vector bundles to a generalized cohomology theory. This ex*
*tension is
discussed in [Ra] and in Shmuel Weinberger's book [W].
The surgery exact sequence becomes:
. .!.Hn+1(P ; L(e)) ! Ln+1(Zss1(P )) ! S(P ) ! Hn(P ; L(e)) ! Ln(Zss1(P )):
Here the homology is homology with coefficients in the nonconnective omega spec*
*trum
based on F=T OP . See [A] for a discussion of generalized homology theories.
We pay a price in that this new version of the structure set may be slightly *
*larger than
the old one. Siebenmann's periodicity map S(M; @M) ! S(M x D4; @) is an isomorp*
*hism
if @M 6= ;, but in the closed case all we know is that there is an exact sequen*
*ce:
0 ! S(M) ! S(M x D4; @) ! Z:
The case M = Sn is a case in which the periodicity map has cokernel Z and the c*
*ase
M = T nis a case in which the periodicity map is an isomorphism. This is discus*
*sed in [N,
p. 81].
Definition 2.4. Let X and Y be spaces and let p : Y ! Z be a map with Z a metr*
*ic
space. A map f : X ! Y is an ffl-equivalence over Z if there exist a map g : Y *
*! X and
homotopies ht : idX ' g O f, kt : idY ' f O g so that the tracks p O f O ht(x) *
*and p O kt(x),
0 t 1, have diameters < ffl.
6
In case X and Y are manifolds and p is the identity map, we have a topologica*
*l rigidity
result.
Theorem (ff-approximation Theorem). If Mn is a topological manifold, n 4, there
is an ffl > 0 so that if f : (N; @N) ! (M; @M) is an ffl-equivalence (over M) w*
*ith f|@N :
@N ! @M a homeomorphism, then f is homotopic (rel boundary) to a homeomorphism.
Remark 2.5. In dimensions 5, this is a theorem of Chapman-Ferry, [ChF]. In dim*
*ension
4 Ferry-Weinberger [FW] prove the same result using work of Quinn [Q]. This res*
*ult also
appears in [Au].
A consequence of this is
Theorem 2.6. If P is a polyhedron and n 5 is given, then there is an ffl > 0 s*
*o that
if f : (N; @N) ! (M; @M) is an ffl-equivalence rel boundary over p : M ! P , th*
*en the
structure (N; f) is in the kernel of the induced map p# : Sn(M) ! Sn(P ).1
The point here is that controlled structures go to controlled structures, i.e*
*., the maps and
homotopies giving the induced structure in Sn(P ) have diameters a predictable *
*constant
multiple of ffl. T. A. Chapman proved the analogous result for simple-homotopy *
*theory in
[Ch].
We will need a result concerning the generalized homology groups Hn(K; L(e)) *
*appearing
in the surgery exact sequence.
Proposition 2.7. For any finite complex K, Hn(K; L(e)) Q ~=Hn+4i(K; ; Q). This
decomposition is natural with respect to maps f : K ! L.
The sum here is over all i, positive and negative. For a finite complex, all*
* but finitely
many of these summands are zero. The proposition follows immediately from the f*
*act that
F=T OP is rationally a product of Eilenberg-MacLane spaces K(Z; 4i). See [Si, x*
*15], [MM,
p. 189].
We will also be using Serre's theory of algebraic topology "mod C." Here is *
*the basic
definition.
Definition 2.8. If C is a class of abelian groups which is closed under the for*
*mation of
subgroups, quotient groups, and group extensions, a homomorphism f : A ! B is s*
*aid to
be a C-monomorphism if ker f 2 C, a C-epimorphism if coker f 2 C, and a C-isomo*
*rphism
if both ker f and coker f are in C.
The basic reference is [S]. See [MT, Chapter 10] for an English summary. Muc*
*h of
algebraic topology and homological algebra works in a mod C setting. In particu*
*lar, we will
use a mod C version of the Five Lemma. The two classes we will be using are the*
* class of
finitely presented groups and the class of torsion groups.
_________________________1
There is a question of whether to write Sn(M) or Sn(M; @M) here. All structure*
*s appearing in this
paper will be "rel @."
7
The actual proof of Theorem 1
For large i, the maps p2 O r : Mi! P2 and ff : P2 ! P1 give us a commuting di*
*agram of
surgery exact sequences.
: :_:_wHn+1(Mi; L(e))L___wn+1(Zss1(Mi))S___wn(Mi)H___wn(Mi;LL(e))n___w(*
*Zss1(Mi))
|u ~=|u |u |u *
* ~=|u
(*) : :_:_wHn+1(P2; L(e))L___wn+1(Zss1(P2))S___wn(P2)H___wn(P2;LL(e))n___w(*
*Zss1(P2))
|u ~|u= |u |u *
* ~=|u
: :_:_wHn+1(P1; L(e))L___wn+1(Zss1(P1))S___wn(P1)H___wn(P1;LL(e))n___w(*
*Zss1(P1))
where the isomorphism of Wall groups comes from the fact that p|Mi induces a ss*
*1-iso-
morphism.
Theorem 2.6 shows that for sufficiently large i, the structure (Mj; fji) is i*
*n the kernel of
(p2 O r)# for all j > i. We will be done if we can show that the kernel of Sn(M*
*i) ! Sn(P2)
is finite, since then two of these structures will be equivalent and Mj will be*
* homeomorphic
to Mj0for some j 6= j0, a contradiction.
Since Wall groups can be infinitely generated, we need to start with the foll*
*owing:
Lemma 2.9. The kernel of (p2 O r)# : Sn(Mi) ! Sn(P2) is finitely generated.
Proof: It follows from the Atiyah-Hirzebruch spectral sequence that Hn(Mi; L(e)*
*) and
Hn(P2; L(e)) are finitely generated. Modulo finitely generated groups, then, a *
*piece of the
diagram (*) becomes:
0 ___wLn+1(Zss1(Mi)) ___wSn(Mi)0___w
~=|u |u
0 ___wLn+1(Zss1(P2)) ___wSn(P2)0___w
so modulo finitely generated groups, (p2 O r)# : Sn(Mi) ! Sn(P2) is an isomorph*
*ism and,
in particular, (p2 O r)# has finitely generated kernel.|
Since ker(p2O r)# is finitely generated, to show that it is finite, it suffic*
*es to show that it
is a torsion group. By property P4 and Proposition 2.7, the homomorphisms (ff O*
* p2 O r)* :
Hm (Mi; L(e))Q ! Hm (P2; L(e))Q and (p2Or)* : Hm (Mi; L(e))Q ! Hm (P1; L(e))Q,
m = n; n + 1, are monomorphisms and Hn+1(Mi; L(e)) Q and Hn+1(P2; L(e)) Q have
the same image in Hn+1(P1; L(e)) Q. Chasing the diagram (*) modulo torsion com*
*pletes
the argument:
If ff 2 Sn(Mi) goes to 0 in Sn(P2), then ff ! 0 in Hn(Mi; L(e)), so ff comes *
*from
ff02 Ln+1(Zss1(Mi)). The image of ff0 in Ln+1(Zss1(P2)), which we will also cal*
*l ff0, comes
from ff002 Hn+1(P2; L(e)). Choose ff(iv)2 Hn+1(Mi; L(e)) so that ff(iv)and ff00*
*have the
same image ff000in Hn+1(P1; L(e)). Then ff(iv)hits ff0 in Ln+1(Zss1(Mi)) and ff*
* is zero by
exactness. Since this was all modulo torsion, this shows that every element of *
*the kernel of
Sn(Mi) ! Sn(P2) is a torsion element.
8
This completes the proof of Theorem 1 for n 5. The extension to the 4-dimens*
*ional
case is accomplished by crossing with a circle, applying the 5-dimensional theo*
*rem, and
splitting back as in [GPW1]. See also [FW, p. 407].|
Remark 2.10. The main result of [GP] says that for any n, the collection of Rie*
*mannian n-
manifolds with sectional curvature bounded below by , diameter bounded above by*
* D, and
volume bounded below by v has a common contractibility function. This set is pr*
*ecompact
by Gromov's Precompactness Theorem, [G], so Theorem 1 shows that this class con*
*tains at
most finitely many homeomorphism types.
There is a sense in which this proof is simpler than the one in [GPW1]. For i*
*nstance, no
use is made of Edwards' disjoint disk theorem or of Quinn's Resolution Theorem.*
* The only
controlled topology we use is our appeal to the ff-Approximation Theorem of [Ch*
*F] (and an
appeal to Quinn's 4-dimensional thin h-cobordism theorem [Q] for the 4-dimensio*
*nal case).|
3. The proofs of properties P1-P4
The proof of P1: The basic reference for these things is [GLP]. An accessible p*
*roof can
also be found in [P2], which contains a survey of the foundations of the subjec*
*t.|
The proof of P2: We begin with a definition.
Definition 3.1. A space X is locally k-connected if for every ffl > 0 there is *
*a ffi > 0 so that
if f : S` ! X is a map, 0 ` k, with diam(f(S`)) < ffi, then there is a map f *
*: D`+1 ! X
with f|S` = f and diam(f(D`+1)) < ffl.
Proposition 3.2. The space X = limMi is locally k-connected for all finite k.
Proof: This is an immediate consequence of Theorem 9 in [P2]. See also [B1].|
Example 3.3. The infinite 1-point union _1i=1Si of i-dimensional spheres Si of *
*diameter
1_
i in `2 is a compact set which is locally k-connected for each k without being*
* locally
contractible. For finite-dimensional spaces, the conditions of "locally k-conne*
*cted for all k"
and "local contractibility" are equivalent.
Given this, the construction of r is relatively straightforward. We show that*
* given n and
ffl > 0 there is ffi > 0 so that if an n-manifold Mi and X are ffi-close in Z, *
*then there is a
map f : Mi ! X which moves points by < ffl in the metric of Z. For polyhedral M*
*i, we
take a fine triangulation and proceed by "local obstruction theory," sending ea*
*ch vertex to
a nearby point of X, using the local contractibility of X to map in the other s*
*implices in
order of increasing dimension. In the general case in which Mi is not polyhedra*
*l, we map
into the nerve of a cover and use the polyhedral argument on the nerve. See [B*
*1] or the
proof of "Main obstruction result" in [P2] for details.
9
That r is a weak homotopy equivalence follows from the same argument using su*
*ccessive
Mi's to verify the homotopy isomorphism on higher and higher homotopy groups. *
*Since
the Mi's are all homotopy equivalent by P3 below, the result follows. |
The proof of P3: This is another direct application of the same argument. This *
*time we
use the contractibility functions on Mi and Mj to build maps both ways and homo*
*topies
from the compositions back to the identity. See [B1] or [P2].
The proof of P4: This is a consequence of Proposition 3.2 and the results of [B*
*2] and
[F1]. In [B2], Borsuk proves:
Theorem (Borsuk). If X is a connected, locally k-connected metric space and X*
* =
lim-{Qi; ffi}, then the system {ss`(Qi); ffi} is stable for ` k and Mittag-Lef*
*fler for ` = k+1.
A system is stable if it is equivalent in some appropriate sense, to a system*
* of isomor-
phisms. A system is Mittag-Leffler if it is equivalent to a sequence of epimor*
*phisms. A
proof of Borsuk's theorem also appears on p. 381 of [F1].
In [F1], the author proves a converse of this theorem up to shape - that a co*
*mpact
metric space which satisfies the algebraic conditions above is shape equivalent*
* to a locally
k-connected compactum. An important step in the argument, which appears on pp. *
*381-
382 of [F1], is to show that if the algebraic conditions are satisfied, then X *
*can be written
as an inverse limit X = lim-{Qi; ffi} of finite CW complexes so that all of the*
* maps ffi are
(k + 1)-connected. That is, it is always possible to choose {Qi; ffi} so that *
*the stability
and Mittag-Leffler properties are exhibited directly. In the present case, wher*
*e X is locally
k-connected for all k, the argument extends without difficulty to give property*
* P4.
This argument is more-or-less independent of the rest of [F1]. The idea is to*
* use Wall's
cell-attaching procedure to start with an arbitrary inverse sequence and progre*
*ssively make
the bonding maps in the system more and more highly connected. The polyhedra P1*
* and
P2 are two stages of the resulting inverse sequence and the map X ! P2 is the n*
*atural
projection from an inverse limit back to any of the finite stages.|
4. Theorems 2 and 3 and a 3-dimensional example
The application of Theorem 1 to proving Theorem 2 was pointed out to the auth*
*or by
Greene and Petersen. In [GrP], Greene and Petersen prove Theorem 2 subject to t*
*he extra
hypothesis that there exist constants C and k 2 (0; 1] so that ae(ffl) Cfflk f*
*or all ffl R. The
authors point out explicitly that the extra hypothesis is needed only to avoid *
*the possibility
of infinite-dimensional limit spaces as constructed in [M]. Since our Theorem 1*
* shows that
the finiteness theorem is valid even in the presence of such infinite-dimension*
*al limits, their
hypothesis is unnecessary.
10
Theorem 3 follows from the proof of Theorem 1 and the arguments of [M]: If f1*
* : M1 ! X
is cell-like, then Theorem 1 of [M] shows that the levels of the mapping cylind*
*er of f give a
path in Mman (n; ae) for some ae. If f2 : M2 ! X is a second cell-like map, we*
* have a second
such path, also approaching X. Of course, the contractibility function will be *
*different, but
we can take the maximum of the two contractibility functions to get a function *
*applying
simultaneously to both paths.
If p2 : X ! P2 is a map as in P4, then for any ffl > 0, we can find a homotop*
*y equivalence
f : M2 ! M1 so that (M2; f) is an ffl-equivalence over p2 O r| : M1 ! P2. Here*
* r is the
mapping cylinder retraction for M(f1). To get smaller and smaller epsilons, we *
*restrict to
levels of the mapping cylinder which are closer and closer to X.
As before, then, infinitely many such Mi's, would lead to an infinite number *
*of elements
of the kernel of Sn(M1) ! Sn(P2) and a contradiction.
Remark 4.1. The author circulated a remarkably short-lived (<24 hours) preprint*
* in which
he claimed to have shown that two manifolds which admit cell-like maps onto the*
* same
compactum must be homeomorphic. The author would like to thank Boris Okun for l*
*ooking
very puzzled when he (the author) began to explain the proof.
Example 4.2. In case there is a counterexample, 3 to the 3-dimensional Poincare*
* Con-
jecture, here is how we construct a counterexample to Theorem 1 in dimension 3.*
* Consider
the sequence {Mi; ci} where Mi= #ij=13 and ci squeezes one summand to a point.
c2 3 3 c3 3 3 3
3 - # - # # - : : :
The inverse limit is an ANR Z. We get a metric space W containing all of the Mi*
*'s together
with Z by taking the inverse limit
id ` c2 a id ` id ` c3a a
3 ----- 3 3#3 -------- 3 3#3 3#3#3- : : :
As in Theorem 1 of [M], the Mi's and Z have a common contractibility function i*
*n this
metric. On the other hand, the existence and uniqueness of prime factorization*
*s for 3-
manifolds shows that the Mi's are not homeomorphic. See the proof of Theorem 2 *
*of [M]
for a similar construction and a picture. An elaboration of this example due to*
* Jakobsche
[J] produces a limit which is homogeneous in the sense that there is a homeomor*
*phism
taking any point to any other point.
11
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