EMBEDDINGS OF HOMOLOGY MANIFOLDS IN CODIMENSION 3 J. L. BRYANT AND W. MIO. 1. Introduction

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1 EMBEDDINGS OF HOMOLOGY MANIFOLDS IN CODIMENSION 3 J. L. BRYANT AND W. MIO 1. Introduction Among the many problems attendant to the discovery of exotic generalized manifolds [2, 3] is the \normal bundle" problem, that is, the classication of neighborhoods of generalized manifolds tamely embedded in generalized manifolds (with the disjoint disks property). In this paper we study the normal structure of tame embeddings of a closed generalized manifold n into topological manifolds V n+q, q 3. If the local index {() 6= 1, then the codimension is necessarily 3 (see, e.g., Proposition 5.4 below). The main result is an extension to ENR homology manifolds of the classication of neighborhoods of locally at embeddings of topological manifolds obtained by Rourke and Sanderson in [10]. We show that for q 3 and n+q 5, germs of tame manifold q-neighborhoods of, or equivalently, controlled homeomorphism classes of (q, 1)-spherical manifold approximate brations over are in one-to-one correspondence with [; BT op q ], where BT op q is the classifying space for stable topological q-microbundle pairs [10]. Manifold approximate brations over topological manifolds have been studied by Hughes, Taylor and Williams in [5]. Our approach is to reduce the study of q-neighborhoods of to the classication of q-neighborhoods of a (stable) regular neighborhood of in euclidean space; this is done using the splitting theorem for manifold approximate brations proved in section 4. As applications, we obtain an analogue of Browder's theorem on smoothings and triangulations of Poincare embeddings (Theorem 5.1) and of the Casson-Haeiger-Sullivan-Wall embedding theorem (Corollary 5.2) for generalized manifolds. These were also obtained independently by Johnston in [6]. 2. Preliminaries A generalized n-manifold is a locally compact euclidean neighborhood retract (ENR) such that for each x 2, ( H k (; r fxg; Z) Z; if k = n, = 0; otherwise. A compact generalized n-manifold is orientable if there is a class 2 H n (; Z) such that the inclusion H n (; Z)! H k (; r fxg; Z) sends to a generator of H k (; r fxg; Z) for every x 2. Achoice of is an orientation for. A subset Y is 1-LCC in Y if, for every x 2 and neighborhood U of x in Y, there is a neighborhood V of x 2 Y such that the inclusion induced homomorphism 1 (V r )! 1 (U r ) is trivial. In codimension 3, we also refer to 1-LCC ENR subsets as tame subsets. Partially supported by NSF grant DMS

2 2 J. L. BRYANT AND W. MIO Given a space, a manifold approximate bration with ber F over is an approximate bration p: M!, where M is a topological manifold and the homotopy berofpis homotopy equivalent tof. (Equivalently, each p,1 (x) has the shape of the space F.) A group G ( 1 (F ) in our constructions) is K-at if Wh(G Z k ) = 0, for every k 0. Let p i : M i!, i 2f0;1g, be continuous maps. A controlled map f c from (M 0! ) to(m 1!) is a proper map f : M 0 [0; 1)! M 1 [0; 1) such that the composition M 0 [0; 1) f M 1 [0; 1) proj M 1 p 1 extends continuously to M 0 [0; 1] via p 0 on M 0 f1g. Similarly, controlled maps f c 0 ;fc 1: M 0! M 1 are controlled homotopic if there is a controlled map H c from M 0 I! to M 1! such that H c jm 0 f0g= f c 0 and Hc jm 1 f1g= f c 1. Controlled homeomorphisms and controlled homotopy equivalences are dened in the obvious way. Remark. (i) This denition is similar to that given in [5], except that we do not require controlled maps and homeomorphisms to be level preserving. However, controlled maps f : M 0 [0; 1)! M 1 [0; 1) are controlled homotopic to levelpreserving mappings through linear homotopies. Similarly, level-preserving controlled homotopic maps are controlled homotopic through level-preserving homotopies. (ii) If M 0 and M 1 are closed manifolds, h is a controlled homeomorphism of the manifold approximate brations p i : M i! with ber F, i 2f0;1g, and 1 (F )is K-at, then an innite sequence of applications of the thin h-cobordism theorem [8] allows us to assume that h preserves a sequence of levels converging to 1, that is, h(m 0 ftg)=m 1 ftg for an innite sequence of t's converging to 1. (iii) When we speak of controlled equivalences f c : M 0! M 1 without specifying the control map p 0 on the domain, it is assumed that p 0 = lim p 1 f t and t!1 that the limit exists, where f t (x) =f(x; t). Although part of our discussion could be carried out in greater generality, unless stated otherwise, we assume that manifold approximate brations p: M! have ber S q,1, q 3, that the total space M is a closed manifold, and that the base space is a closed ENR homology manifold. Let p: M! be a manifold approximate bration. A controlled structure on p is a controlled homotopy equivalence f c : N! M, where N! is a manifold approximate bration. The controlled structure set of p, Sc(p), is the collection of all controlled homeomorphism classes of controlled structures on p: M!. For computational purposes, we next identify Sc(p) with a certain bounded structure set, in the sense of Ferry and Pedersen [4]. Given p: M!, assume that is tamely embedded in S N, N large, and that is given the induced metric. Let O() denote the open cone on, and let }: M [0; 1)! O() be dened by ( (p(m);t); if t>0; }(m; t) = 0; if t =0,

3 EMBEDDINGS OF HOMOLOGY MANIFOLDS 3 where 0 denotes the cone point. We wish to identify Sc(p) and S >0 b (}), where S >0 b (}) is the bounded structure set of }: M [0; 1)! O() away from zero (see [4] for more details). Let :[0; 1)! [0; 1) be a homeomorphism. A radial reparametrization (via ) of a controlled structure f c on p: M! may fail to yield a bounded structure (away from 0) on }, ifconvergence near is too slow. This can be corrected with a suitable radial contraction of the given structure, as follows. Let : [0;1)! [0; 1) be a homeomorphism such that (x) x, for every x, and let f c be a controlled structure on p: M! represented, say, by the levelpreserving map f : M 0 [0; 1)! M [0; 1). The -contraction of f c is the controlled structure represented by the composition M 0 [0; 1) id,1 M 0 [0; 1) f M [0; 1) id M [0; 1): If the contraction is ne enough, then its radial reparametrization under gives a bounded structure on }: M [0; 1)! O() away from 0. Appropriate restrictions on admissible functions guarantee that controlled homeomorphic structures are mapped to equivalent bounded structures, thus dening a map Sc(p)! S >0 b (}). Conversely, if f 0 : N 0! M[0; 1) represents a bounded structure away from zero, then N 0 has a simply-connected, tame end with respect to the control map : N 0! given by the composition N 0 f 0 M [0; 1) proj M p : By the end theorem [8] we can assume that, in a neighborhood of the end, N 0 = N [0; 1), and that the maps t : N! given by t (x) =(x; t) converge to a spherical manifold approximate bration 1 : N!, as t! 1. Moreover, 1 : N! is controlled homotopy equivalent top:m!, under the map induced by f 0. This establishes a one-to-one correspondence Sc 0 M # A,! >0 S b between controlled and bounded structure sets. M [0; 1) # O() 1 A 3. Local structure Let n be a closed oriented generalized n-manifold and V n+q a topological manifold, q 3. If is 1-LCC in V and q 3, then has a mapping cylinder neighborhood E = C p, where is a manifold approximate bration with homotopy bers q,1 [8, 12]. Moreover, this spherical manifold approximate bration structure is well dened up to controlled homeomorphismsover. Conversely, by Proposition 3.1 below, any such spherical manifold approximate bration arises as the normal structure of a tame embedding of. This result is a consequence of Edwards-Quinn's characterization of manifolds, and is proven in [8] for polyhedral homology manifolds. Proposition 3.1. If p is a manifold approximate bration with homotopy ber S q,1, q 3, then the mapping cylinder E = C p of p is a topological manifold, provided that n + q 5. Furthermore, is tamely embedded ineas the zero section.

4 H H 4 J. L. BRYANT AND W. MIO Classifying manifold neighborhoods of is, therefore, equivalent to classifying spherical manifold approximate brations over, up to controlled homeomorphisms. We rst address this problem within a xed controlled homotopytype over. Let be a manifold approximate bration with homotopy bers q,1, and let p 1 1! be a manifold approximate bration controlled equivalent to pvia a level-preserving controlled homotopy 1 [0; 1) [0; 1) : p The map induces a homotopy equivalence ~ :(E 1 ;@E 1 )! where E 1 and E are the mapping cylinders of p 1 and p, respectively. Let ( ~ ) 2 [E; G=Top] = [; G=Top] = H n (; G=Top) be the normal invariant of ~. Normal invariants of mapping cylinders induce a map : # 1 A,! Hn (; G=Top) given by ([ c ]) = ( ~ ), for any controlled structure c. Proposition 3.2. is a bijection. Proof. The result follows from a comparison of the controlled surgery exact sequence of with the G=Top-homology Gysin sequence of It is a consequence of the 5-lemma applied to the commutative diagram H n+q (; L) = H n+q,1 (@E; G=Top) H n+q,1 (; L) id = H n+q (E; G=Top) n (; G=Top) n+q,1 (@E; G=Top) n+q,1 (E; G=Top) The rst row is the bounded surgery sequence away from 0 [0; 1)! O(), under the identication of structure sets described in section 2. The second row is the G=Top-homology exact sequence of the pair with the term H n+q G=Top) identied with [E; G=Top] = [; G=Top] = H n (; G=Top), via Poincare duality. For k 1, the isomorphism H n+k (; L) = H n+k (; G=Top) follows from the Atiyah-Hirzebruch spectral sequence. Wenow consider the general classication problem. Let Nq() be the collection of germs of tame codimension q manifold neighborhoods of. Twoembeddings k : n! Vk n+q, k 2f1;2g, represent the same element of Nq(), if there are neighborhoods N k of in V k, and a homeomorphism h: N 1! N 2 such that h 1 = 2. Our previous discussion shows that Nq() is in 1{1 correspondence with controlled homeomorphism classes of (q, 1)-spherical manifold approximate brations over. Let BT op q+k;k denote the classifying space for topological microbundle pairs k k+q, where k denotes the trivial microbundle of rank k. In [10], Rourke and

5 EMBEDDINGS OF HOMOLOGY MANIFOLDS 5 Sanderson showed that if M is a topological manifold, there is a bijection : Nq(M)! [M; BT op q ]; where BT op q = lim k!1 BT op q+k;k.tothe embedding M V, they associate the pair M M V j M M, which represents the \formal" normal bundle of M in V. Here, M is a stable inverse to M. A closed generalized manifold has a canonical (up to controlled homeomorphisms) stable normal spherical manifold approximate bration structure on the Spivak normal bration determined by neighborhoods of embeddings of in large euclidean spaces; we shall denote it. The uniqueness of this stable structure follows from the relative end theorem and the thin h-cobordism theorem [8] applied to a concordance between any twoembeddings of in R N, N large. On the other hand, Ferry and Pedersen have shown that the Spivak normal bration of an ENR homology manifold has a canonical Top reduction fp [4]. Notice that, if the local index of 6= 1,we cannot have a simultaneous geometric realization of both structures since the index is multiplicative. In other words, we cannot have atop-bundle over whose total space is a manifold. Our approach is to use both and fp to reduce the study of q-neighborhoods of to the study of q-neighborhoods of a (stable) neighborhood of in euclidean space. The fact that there is a bijective correspondence between these is stated as Corollary 3.4. We begin by dening : Nq()! [; BT op q ], for any closed generalized manifold. Let n V n+q be a tame embedding, q 3, and let E = C p be a mapping cylinder neighborhood of with projection p: E!. Let W be the total space of = p ( fp ). Since has the same controlled homotopy type as fp, and j = fp, the splitting theorem proved in the next section shows that is controlled homeomorphic to an approximate bration that restricts to over. Moreover, the mapping cylinder N of the projection of is embedded in W as a locally at submanifold. N W E Figure 3:1 A relative version of this construction shows that any two such splittings are concordant. Therefore, the assignment V 7! N j N j W j N j induces a classifying map : Nq()! [; BT op q ], which coincides with the Rourke- Sanderson map when is a topological manifold. Theorem 3.3. Let n be a closed generalized manifold. The map : Nq()! [; BT op q ] is a bijection, provided that q 3 and n + q 5. Using the same notation, dene }: Nq()! Nq(N) by associating to V the q-neighborhood N W. Corollary 3.4. }: Nq()! Nq(N) is a bijection.

6 [N; E 6 J. L. BRYANT AND W. MIO Proof. Neighborhoods of N are classied by 1 : Nq(N)! [N; BT op q ], where 1 ([ V ]) is represented by the microbundle pair N N W j N N [10]. Therefore, there is a commutative diagram Nq() } Nq(N) [; BT op q ] The result follows from Theorem 3.3. = 1 BT op q ]: Proof of Theorem 3.3. is injective. Let p i i!, i 2 f0;1g,be(q,1)-spherical manifold approximate brations such that ( E 0 )=(E 1 ). Adding subscripts to the notation introduced above (see gure 3.1), Theorem 3.2b of [10] implies that W 0 and W 1 are equivalent neighborhoods of N, i.e., there is a homeomorphism H : W 1! W 0 inducing the identity onn. Analogous to BT op q, there is a classifying space BG q for pairs of spherical brations [10]. Since such pairs split uniquely, BG q is homotopy equivalent to the classifying space for spherical brations BG q. Since controlled homotopy classes of approximate brations are in one-to-one correspondence with ber homotopy classes of spherical brations under the path bration map [5], the fact that the image of ( E 0 ) and ( E 1 ) are the same under the forgetful map BT op q! BG q implies that H induces a controlled homotopy equivalence f c 0 over, which gives a homotopy equivalence f ~ : E 1! E 0 of mapping cylinders. By Proposition 3.2, to establish the injectivity of, it suces to show that the normal invariant of f ~ is trivial. Since f ~ restricts to the identity on, and W i is the total space of the bundle over E i obtained as the pull-back of fp under the projection p i : E i!, there is a bundle map F : W 1! W 0 W 1 F W 0 p 1 E 1 ~ f p 0 0 : covering ~ f : E 1! E 0. Therefore, the normal invariants (F ) 2 [W 0 ; G=Top] = [; G=Top] and ( ~ f) 2 [E 0 ; G=Top] = [; G=Top] are the same. Since F can be assumed to be homotopic to H as maps of pairs, it follows that ( ~ f)=(f)=0. is surjective. Let the microbundle pair k q+k represent a given element 2 [; BT op q ]. Let be the (q, 1)-spherical bration underlying, and let E be the mapping cylinder of p. Abusing notation, the natural projection p: E! gives (E;@E) the structure of an arbitrarily ne Poincare space over. Furthermore, as stable spherical brations (3.1) fp = sp E j ; where E sp is the Spivak normal bration of E. Therefore, equation 3.1 determines a Top reduction of E sp j. Since E deformation retracts to, we also obtain a Top

7 EMBEDDINGS OF HOMOLOGY MANIFOLDS 7 reduction of sp E. Let (E;@E) p be a surgery problem associated to this reduction. Crossing with R, we obtain a bounded surgery problem ( ~ M;@ ~ M) ~ (ER;@ER) p 0 O( + ) where + = qfagis a disjoint union, and in \polar coordinates" in O( + ), ( p 0 (x; t); if t>0; (x; t) = (a; jtj); if t 0. By the bounded, theorem [4], we can assume that ~ is a bounded homotopy equivalence. Now, we split this equivalence near 1 to obtain a manifold approximate bration over. ~ M has a tame end (near +1) with respect to the ~ M R p ; we can assume by the end theorem that in a neighborhood of the ~ M ~ E 1 [0; 1). For each 0<t<1, let p t 1! be the 1 ftg proj Then, p 1 = lim t!1 p t 1! is a manifold approximate bration such that the spherical brations underlying ( E 1 ) and are the same. The stability theorem for G q =T op q [10], q 3, gives a pull-back diagram E BT op q BTop p : BG q BG: Hence, the dierence between and ( E 1 ) is stable, and denes an element 2 [; G=Top] = H n (; G=Top). Proposition 3.2 applied to p 1 : E 1! gives a manifold approximate and a controlled equivalence 1 such that ([ c ]) =. Then, ( E) =. This concludes the proof. Corollary 3.5. Let n be a closed generalized manifold. If q 3 and n + q 5, (q, 1)-spherical manifold approximate brations over are classied bybtop q, i.e., there is a one-to-one correspondence between controlled homeomorphism classes of (q, 1)-spherical manifold approximate brations over and [; BT op q ].

8 8 J. L. BRYANT AND W. MIO Since BG q! BG q is a homotopy equivalence and G q =T op q is stable when q 3, the classication of spherical manifold approximate bration structures obtained in Proposition 3.2 can be rephrased in terms of reductions of structural groups, as follows. Corollary 3.6. Let be a closed generalized n-manifold, and let be a(q,1)- spherical bration over. If q 3 and n + q 5, then manifold approximate brations over ber homotopy equivalent to are in 1{1 correspondence with ber homotopy classes of lifts to BT op q of the map! BG q that classies. Remark. The classication of manifold approximate brations with spherical bers obtained in Corollary 3.5, the H-space structure on BTop induced by Whitney sums of bundles, and the stability theorem for G q =T op q, for q 3, suggest possible denitions of Whitney sums and pull-backs of spherical manifold approximate brations over ENR homology manifolds. However, such denitions do not seem to be entirely satisfactory. For example, one would not obtain product formulae for characteristic classes, since 0-dimensional classes (like the local index of a homology manifold) are not visible to the operation in BTop arising from Whitney sums of bundles. In order to account for these, it appears to be necessary to address the more general classication problem of tame neighborhoods of generalized manifolds in generalized manifolds with the disjoint disks property. We conjecture that these q-neighborhoods are classied by BT op q Z. From the viewpoint ofthe techniques utilized in this paper, the main obstacles to completing such a study are the validity of the s-cobordism theorem and of the simply-connected end theorem for generalized manifolds with the disjoint disks property. 4. A splitting theorem Let E m be a compact topological m-manifold, and p: M m+r! E be a manifold approximate bration with homotopy ber F, where F is a closed r-manifold. For the duration of this section we adopt the following notation: if A E, then ^A = p,1 (A) M. Let n E m be a closed ENR homology manifold tamely embedded in E, and let q : ^! denote the restriction of p to ^. In this generality, ^ is not necessarily an ANR, and q may not be an approximate bration. Denition 4.1. p: M! E is split along, if ^is a closed (n + r)-manifold tamely embedded in M, and q = pj ^ : ^! is an approximate bration. Suppose q 1 : N! is an approximate bration with homotopy berf, where N is a closed (n + r)-manifold Theorem 4.2. If q 1 : N! is berwise shape equivalent to q : ^! over, m, n 3, r 3, n + r 5, and 1 (F ) is K-at, then p: M! E is controlled homeomorphic to an approximate bration p 1 : M! E, which is split along and restricts to q 1 over. Proof. Let V be a mapping cylinder neighborhood of in E with mapping cylinder projection : V!. Then int ^V has a tame end (the control map being the composition of p with projection on a collar ). Since 1 (F )=0, ^V has a controlled collar at 1 [8]. Let U be a compact manifold in int ^V containing ^ obtained by removing a small open collar from the end of ^V. Then

9 EMBEDDINGS OF HOMOLOGY MANIFOLDS 9 the inclusion ^ U is a shape equivalence. The map p: U r ^! also has a tame end, hence a controlled collar at 1. Thus U has a \mapping-cylinder-like" structure over ^, controlled over, in the sense that small neighborhoods of ^ in U can be isotopied arbitrarily close to ^ by isotopies that x ^ and are controlled over. (That is, ^ is a tame FANR in U over.) Since the manifold approximate bration q 1 : N! is berwise shape equivalent top: ^!, there is a controlled homotopy equivalence G c : N! U over. Let G: N [0; 1)! U [0; 1) represent G c. As in section 2, under an appropriate radial reparametrization, we may assume that G is a bounded homotopy equivalence over O(), the open cone on. By the bounded analogue of the Casson-Haeiger- Sullivan-Wall embedding theorem [11], after a bounded homotopy over O(), we can assume that G is an embedding. Pushing the image of G toward ^ using the mapping-cylinder-like structure on U, wemay also assume that G(N ftg) ^V t, where V t V is the part of the mapping cylinder having mapping cylinder parameter t. Set Z = G(N [0; 1) [ ^ f1g) U[0; 1]. Then U [0; 1]rZ has a tame end over, hence, a controlled collar over. The closed region between a boundary of the end [0; 1] is a thin h-cobordism of triples (over ), hence, a product. Thus, there is a controlled homeomorphism ([0; 1]; 0; 1) [0; 1)! U ([0; 1]; 0; 1) r Z over, with j@u [0; 1] [0;] = id for some >0. It is not dicult to show that (@U [0; 1]ftg)isan(t)-thin h-cobordism over, where (t)! 0ast!1. Thus, an innite sequence of applications of the thin h-cobordism theorem yields a homeomorphism h: (Ur^)[0; 1]! (U [0; 1]rZ), controlled over. The composition pprojh,1 :(U[0; 1]rZ)! V r extends to a map H : U [0; 1]! V. Since h is the identity in a neighborhood H can be used to deform p to an approximate bration p 1 that agrees with p outside U and with q 1 on N = G(N f0g), and is controlled homeomorphic to p. 5. Taming Poincare embeddings Let n be a closed generalized manifold and V n+q a compact topological manifold. Following [1] (see also [7, 11]), we dene a Poincare embedding of in V to be a triple consisting of (i) a (q, 1)-spherical bration over, with projection ; (ii) a nite Poincare pair (iii) a (simple) homotopy equivalence h: C [ E! V, where E is the mapping cylinder of p, and C \ E Remark. 1. If is tamely embedded in V, let the spherical manifold approximate - bration represent the normal structure to. Under the path bration construction, this approximate bration determines a spherical bration over which is controlled homotopy equivalent to p:@e!. Thus, underlying any tame embedding, there is a Poincare embedding of in V. 2. Lemma 11.1 of [11] shows that (ii) follows from (iii), when q 3. As in smoothing theory, we rst consider reductions of the structural group of p: E! to T op q. GivenaPoincare embedding h: C E! V, let f s be the

10 10 J. L. BRYANT AND W. MIO composition i E C E h V i V R k ; k large. By general position, we can assume that f s is a tame embedding. Since any other embedding homotopic to f s is concordant tof s, the stable controlled homeomorphism type of the normal spherical manifold approximate bration is well-dened. Thus, associated to a Poincare embedding h: C E! V, there is a natural stable Top-reduction The stability theorem [10] for G q =T op q, q 3, implies the same unstably, that is, associated to a Poincare embedding, there is a canonical T op q -reduction We now state the extension of Browder's \Top-Hat" theorem to embeddings of ENR homology manifolds into topological manifolds (see also [6]). For smooth or PL manifolds, a proof of this result is given in [11]. Theorem 5.1. Let (;(C; E);h) be a Poincare embedding of a closed generalized manifold n into a compact topological manifold V n+q, with q 3 and n + q 5. Then, there is a tame embedding of in V inducing the given Poincare embedding. Proof. The proof follows much the same line as the proof of Theorem 11.3 of [11]. Let represent the canonical Top q -reduction is a (q, 1)-spherical manifold approximate bration. Since c induces a simple homotopy equivalence : ~ E! E, wemayassume that E = E and h: C [@E E! V. Let g : V! C E be a homotopy inverse to h. After removing a small open collar from E, we can assume that g is transverse Let A = g,1 (E). Then gja: is a degree 1 normal with normal invariant 2 [ E; G=Top] = [ ; G=Top ] = H n (; G=Top). Proposition 3.2 implies that there is a manifold approximate bration p 1 1! and a controlled homotopy equivalence c such that ( ~) =. Hence, there is a normal bordism F 1 :(U; U 0 )! between g and ~: (E 1 ;@E 1 Identify A U with A f1gv f1gv I to obtain W =(VI)[ A U. There is a degree-one normal map F :(W;Vf0g;@ + W)! (C[EI;C[Ef0g;C[Ef1g) inducing h on V f0g and ~ on E + W.By the, theorem we can do surgery on W rel V f0g[e 1 to get an s-cobordism W 0 between V f0gand V 0, with E 2 V 0. By the s-cobordism theorem W 0 = V I; hence, we get an embedding f : E 1,! V realizing the given Poincare embedding. Notice that 1! arises as the normal structure to under the embedding 1! is controlled homeomorphic Corollary 5.2. Suppose n is a closed generalized n-manifold, V n+q is a compact topological (n + q)-manifold, (n + q) 5, q 3, and f :! V is a homotopy equivalence. Then f is homotopic to a tame embedding. Proof. Identical to the proof of Corollary of [11]. Corollary 5.3. Suppose that is a closed generalized n-manifold, n 5. Then there is a tame embedding of into a topological manifold of dimension n +3. Proof. By [4], the Spivak normal bration of admits a Top-reduction, which gives a degree-one normal map f : M!, where M is a topological n-manifold. By the, theorem, we can do surgery on f id: M B 3! B 3, to get a (simple)

11 EMBEDDINGS OF HOMOLOGY MANIFOLDS 11 homotopy equivalence F )! ( B 3 ; S 2 ). Apply 5.2 to a homotopy inverse of F restricted to. In contrast to 5.3 we have the following well-known fact. Proposition 5.4. If is a closed generalized n-manifold and () 6= 1, then there is no compact topological manifold ) controlled homotopy equivalent to ( B 2 ;S 1 ) over. Proof. If there were, the innite cyclic cover U corresponding to the Z-factor of 1 (@V )would have a tame end over. The end theorem would then produce a completion U of the end over, hence, a cell-like that is, a resolution of. But this would imply that () = 1[9]. References [1] W. Browder, Embedding smooth manifolds, Proc. Internat. Cong. Mathematicians, Moscow (1968), 712{719. [2] J. Bryant, S. Ferry, W. Mio and S. Weinberger, Topology of homology manifolds, Bull. Amer. Math. Soc. 28 (1993), 324{328. [3], Topology of homology manifolds, Ann. of Math. 143 (1996), 435{467. [4] S. C. Ferry and E. K. Pedersen, Epsilon surgery theory, in Novikov conjectures, index theorems and rigidity,vol. 2 (Oberwolfach, 1993), 167{226, London Math. Soc. Lecture Ser., 227, Cambridge University Press, Cambridge, [5] C. B. Hughes, L. R. Taylor and E. B. Williams, Bundle theories for topological manifolds, Trans. Amer. Math. Soc. 319 (1990), 1{65. [6] H. Johnston, Transversality for non-manifolds, Thesis, The University of Chicago, [7] N. Levitt, On the structure of Poincare duality spaces, Topology 7 (1968), 369{388. [8] F. Quinn, Ends of maps I, Ann. of Math. 110 (1979), 275{331. [9], An obstruction to the resolution of homology manifolds, Michigan Math. J. 301 (1987), 285{292. [10] C. P. Rourke and B. J. Sanderson, On topological neighborhoods, Compositio Math. 22 (1970), 387{424. [11] C. T. C. Wall, Surgery on compact manifolds, Academic Press, [12] J. E. West, Mapping Hilbert cube manifolds to ANR's: a solution to a conjecture of Borsuk, Ann. of Math. 106 (1977), 1{18. Department of Mathematics, Florida State University, Tallahassee, FL address, J. L. Bryant: bryant@math.fsu.edu address, W. Mio: mio@math.fsu.edu