Topological finiteness theorems for manifolds
in Gromov-Hausdorff space
StevenC. Ferry
SUNY at Binghamton
April 26, 1993
Abstract. We give general conditions under which precompact sets of topolo*
*gicalmani-
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 andcontrolled topol*
*ogy 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*
*b ound
on volume, and upper bound ondiameter. One ofthe key steps in their argument i*
*s 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 metricball 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 andthe
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 Hausdorffdistance*
* between
!! X andY is
!
!! dHZ(X;Y ) = inffffl > 0 j Xae Nffl(Y );Y ae Nffl(X)g:
!
!!(ii)If X and Y are compact metric spaces,the Gromov-Hausdorffdistance from X *
*to Y
!! is
!! dGH (X; Y) = infZfdHZ(X;Y)g
!
Partially supported by NSFGrant DMS 9003746. The author would also like to than*
*k the University
of Chicago and the Institute for Advanced Study for support while some of this *
*work was accomplished.
Typeset by LAMS-T*
*EX
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.
Definition1.3. Let Mman (n; ae) be the set of all(X; d) 2 C Msuch 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 theoremin Mman (n;ae).
Theorem 1. If n 6= 3and C is a subset of Mman (n; ae) such that the closure of*
* C is compact
in CM, then Ccontains only finitely many homeomorphism types of topological man*
*ifolds.
It is well-known [HM, KS]that in dimensions 5 there are at most finitely many*
* smooth
manifolds homeomorphic to a given top ological manifold, so we have:
Corollary 1.4. If n 5 and Cis a subset of Mman (n;ae) whose elements are smoo*
*th
manifolds and the closure of C is compact in C M, thenC contains only finitely *
*many
diffeomorphismtypes.
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, t*
*he
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 a*
*n example
of Dranishnikov [D]to show that a sequence of S7's in Mman(7; ae) can con*
*verge to
an infinite-dimensional limit.
(iii)Such behavior cannot occur in the presence of a lower bound on sectional*
* curvature,
so the validity of TheoremA of [GPW1] is not affected by this error. See *
*[GPW2]
for details.
A second finiteness theoremfollows from our Theorem 1 and work of Greene-Pete*
*rsen
[GrP].
Theorem 2. If n 4 andC (ae; V0; n) is the subset of Mman (n; ae) consisting of*
* Riemannian
manifolds such that