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 Bibliography [A] J. F. Adams, Stable homotopy and generalised homology, U. of Chicago Press,* * Chicago, 1974. [Au] K. K. Au, Thesis, University of California at San Diego, 1990. [B1] K. Borsuk, On some metrizations of the hyperspace of compact sets, Fund. M* *ath. 41 (1955), 168-201. [B2] _______, Theory of shape, Monografie Mat. vol. 59, Polish Science Publ., W* *arszawa, 1975. [Ch] T. A. Chapman, Invariance of Torsion and the Wall Finiteness Obstruction, * *Canad. J. Math. 326 (1980), 583-607. [ChF] T. A. Chapman and S.C. Ferry, Approximating homotopy equivalences by home* *omorphisms, Amer. J. Math. 101 (1979), 583-607. [D] A. N. Dranishnikov, On a problem of P.S. Alexandroff, Matematicheskii Sborn* *ik 135 (1988). [F1] S. Ferry, A stable converse to the Vietoris-Smale theorem with application* *s to shape theory, Trans. Amer. Math. Soc. 261 (1980), 369-386. [F2] _______, Counting simple-homotopy types in Gromov-Hausdorff space, preprin* *t. [FW] S. Ferry and S. Weinberger, Curvature, tangentiality, and controlled topol* *ogy, Invent. Math. 105 (1991), 401-414. [GrP] R. Greene and P. Petersen, Little topology, big volume, Duke Math. J. 67 * *(1992), 273-290. [GLP] M. Gromov, J. Lafontaine, P. Pansu, Structures metrique pour les varietes* * riemanniennes, Cedic/Fernand Nathan, 1981. [G] K. Grove, Metric Differential Geometry, in Differential Geometry, Proc. Nor* *dic Summer School, Lyngby 1985, (ed. V.L. Hansen), Springer Lecture Notes in Math. 1263 (1987), 171-2* *27. [GP] K. Grove and P. Petersen, Bounding homotopy types by geometry, Annals of M* *ath. 128 (1988), 195-206. [GPW1] K. Grove, P. Petersen and J. Wu, Geometric finiteness theorems via contr* *olled topology, Inventiones Mathematicae 99 (1990), 205-213. [GPW2] _______, Correction to geometric finiteness theorems in controlled topol* *ogy, Inventiones Mathemat- icae 104 (1991), 221-222. [HM] M. Hirsch and B. Mazur, Smoothings of piecewise-linear manifolds, Annals o* *f Math. Studies 80, Princeton University Press, Princeton, NJ. [J] W. Jakobsche, The Bing-Borsuk Conjecture is stronger than the Poincare Conj* *ecture, Fund. Math. CVI (1980), 125-134. [KS] R. Kirby and L. C. Siebenmann, Foundational essays on topological manifold* *s, smoothings, and trian- gulations, Princeton University Press, Princeton, New Jersey, 1977. [MM] I. Madsen and J. Milgram, The classifying spaces for surgery and cobordism* *, Annals of Math. Studies 92, Princeton University Press, Princeton, NJ. [M] T. Engel Moore, Gromov-Hausdorff convergence to non-manifolds, Journal of G* *eometric Analysis (to appear). [MT] R. Mosher and M. Tangora, Cohomology operations and applications of homoto* *py theory, Harper and Row, New York, 1968. [N] A. Nicas, Induction theorems for groups of manifold homotopy structure sets* *, Memoirs AMS 267 (1982). [P1] P. Petersen V, A finiteness theorem for metric spaces, Journal of Differen* *tial Geometry 31 (1990), 387-395. [P2] _______, Gromov-Hausdorff convergence of metric spaces, Proc. Symposia in * *Pure Math., 1990 Sum- mer institute on Differential Geometry (to appear). [Q] F. Quinn, Ends of maps, III, J. Differ. Geom. 17 (1982), 503-521. [Ra] A. A. Ranicki, The total surgery obstruction, Algebraic topology Arhus 197* *8, SLN #763, Springer Verlag, Berlin, 1979, pp. 275-316. [S] J. P. Serre, Groupes d'homotopie et classes de groupes abelienes, Ann. Math* *. 58 (1953), 258-294. [Si] L. C. Siebenmann, Topological manifolds, Proceedings of the International * *Congress of Mathemati- cians, Nice, Gauthier-Villars, Paris, 1971, pp. 133-163. [Wa] C.T.C. Wall, Surgery on Compact Manifolds, Academic Press, New York, 1971. [Walsh] J. Walsh, Dimension, cohomological dimension, and cell-like mappings, S* *hape Theory and geometric topology, SLN 870, Springer Verlag, 1981, pp. 105-118. [W] S. Weinberger, The topological classification of stratified spaces, U. of C* *hicago Press, Chicago (to appear).