POINCARE DUALITY SPACES John R. Klein Fakult"at f"ur Mathematik, Universit"at Bielefeld 33615 Bielefeld, Germany Introduction At the end of the last century, Poincare discovered that the Betti numbers of a closed oriented triangulated topological n-manifold Xn bi(X) := dimR Hi(X; R) satisfy the relation bi(X) = bn-i(X) (see e.g., [Di, pp. 21-22]). In modern language, we would say that there exists a chain map C*(X) -! Cn-* (X) which in every degree induces an isomorphism H*(X) ~=Hn-* (X) : The original proof used the dual cell decomposition of the triangulation of X. As algebraic topology developed in the course of the century, it became possible to extend the Poincare duality theorem to non-triangulable topological manifolds, and also to homology manifolds. In 1961, Browder [Br1] proved that a finite H-space satisfies Poincare duali* *ty. This result led him to question whether or not every finite H-space has the homotopy type of a closed smooth (= differentiable) manifold. Abstracting further, one asks: Which finite complexes have the homotopy type of closed topological manifolds? of closed smooth manifolds? To give these questions more perspective, recall that Milnor had already shown in 1956 that there exist several distinct smooth structures on the 7- sphere [Mi1]. Furthermore, Kervaire [Ke] constructed a 10-dimensional P L manifold with no smooth structure. It is therefore necessary to distinguish between the homotopy types of topological and smooth manifolds. Kervaire and Milnor [K-M] systematically studied groups of the h-cobordism classes of homotopy spheres, where the group structure is induced by connected sum. They showed that these groups are always finite. In dimensions 5 the h- cobordism equivalence relation is just diffeomorphism, by Smale's h-cobordism theorem [Sm]). 1 2 JOHN R. KLEIN Since topological manifolds satisfy Poincare duality (with respect to suitab* *le coefficients), the existence of a Poincare duality isomorphism is a necessary condition for a space to have the homotopy type of a closed manifold. Such a space is called a Poincare duality space, or a Poincare complex for a finite CW complex. Poincare complexes were to play a crucial role in the Browder-Novikov- Sullivan-Wall surgery theory classification of manifolds. We can view the surgery machine as a kind of descent theory for the forgetful functor from manifolds to Poincare complexes: o Given a problem involving manifolds, it is often the case that it has an analogue in the Poincare category. o One then tries to solve the problem in the Poincare category, where there is more freedom. In the latter, one has techniques (e.g., homotopy theory) that weren't available to begin with. o Supposing that there is a solution to the problem in the Poincare cate- gory, the last step is to lift it back to a manifold solution. It is he* *re that the surgery machine applies. Except in low (co)dimensions, the only obstruction to finding the lifting is given by the triviality of a cert* *ain element of an L-group Ln(ss). Thus surgery theory gives an approach for solving manifold classification prob- lems, modulo the solution of the corresponding problem for Poincare complexes. In general, a Poincare duality space is not homotopy equivalent to a topo- logical manifold. Thus Poincare duality spaces fall into more homotopy types than topological manifolds. In 1965, Gitler and Stasheff [G-S] constructed an example of a simply connected finite complex X which satisfies 5-dimensional Poincare duality, but which isn't the homotopy type of a closed topological manifold. This example has the homotopy type of a complex of the form (S2 _ S3) [ e5, with respect to a suitable attaching map S4 -! S2 _ S3. More specifically, X is the total space of a spherical fibration S2 -! X -! S3 which admits a section. By the clutching construction, such a fibration is classified by an element of ss2(Aut *(S2)) ~=ss4(S2) = Z=2. We take X to correspond to the generator of this group. Returning to Browder's original question about finite H-spaces, it is worth remarking that at the present time there is no known example of a finite H- space which isn't the homotopy type of a closed smooth manifold. Outline. x1 concerns homology manifolds, which are mentioned more-or-less for their historical interest. In x2 we define Poincare complexes, following Wall. I then mention the various ways Poincare complexes can arise. x3 is an ode to the Spivak normal fibration. I give two proofs of its existence. The first essentially follows Spivak, and the second is due to me (probably). In x4 I outline some classification results about Poincare complexes in low dimensions, and I also give an outline as to what happens in general dimensions in the high* *ly connected case. In x5 I describe some results in Poincare embedding theory and further connections to embeddings of manifolds. x6 is a (slightly impious) discussion of the Poincare surgery programs which have been on the market for the last twenty five years or so. I've also included a short appendix on the POINCARE DUALITY SPACES 3 status of the finite H-space problem. The bibliography has been extended to include related works not mentioned in the text. Acknowledgement. I am much indebted to Andrew Ranicki for help in research- ing this paper. Thanks also to Teimuraz Pirashvili for help with translation from the Russian. 1. Forerunners of Poincare duality spaces Spaces having the homological properties of manifolds have a history which dates back to the 1930s, and are to be found in the work of Cech, Lefschetz, Alexandroff, Wilder, Pontryagin, Smith and Begle. These `generalized n- manifolds' (nowadays called homology manifolds) were defined using the local homology structure at a point. The philosophy at the time of their introduc- tion was that these spaces were supposedly easier to work with than smooth or combinatorial manifolds. We recall the following very special case of the definition (for the general definition and the relevant historical background see [Di, pp 210-213]). An (ANR) homology n-manifold X is a compact ANR with local homology groups aeZ if* = n H*(X; X \ {x}) = H*(Rn; Rn \ {0}) = (x 2 X) : 0 if* 6= 0 Now, if our ultimate goal is to study the homotopy properties of manifolds, this definition has an obvious disadvantage: it isn't homotopy invariant. It is easy to construct a homotopy equivalence of spaces X -'! Y such that X is a homology manifold but Y is not a homology manifold. The notion of Poincare duality space is homotopy invariant, offering a remedy for the problem by ignoring the local homology structure at each point. Any space homotopy equivalent to a Poincare duality space is a Poincare duality space. 2. The definitions There are several different flavors of Poincare complex in the literature [W* *a3], [Wa4], [Le1], [Spi]. We shall be using Wall's definition in the finite case, wi* *thout a Whitehead torsion restriction. Suppose that X is a connected finite CW complex whose fundamental group ss = ss1(X) comes equipped with a homomorphism w :ss -! { 1}, which we shall call an orientation character. Let = Z[ss] denote the integral group rin* *g. Define an involution on by the correspondence g 7! g, where g = w(g)g-1 for g 2 ss. This involution will enable us to convert right modules to left modules and vice-versa. For a right module M, let w M denote the corresponding left module. For a left module N, we let Nw denote the corresponding right module. Let C*(Xe) denote the cellular chain complex of the universal covering space Xe of X. Since ss acts on Xe by means of deck transformations, it follows that C*(Xe) is a (finitely generated, free) chain complex of right -modules. For a right -module M, we may therefore define H*(X; M) := H-* (Hom (C*(Xe); M)) H*(X; M) := H*(C*(Xe) w M) : 4 JOHN R. KLEIN Given another right -module N, and a class [X] 2 Hn(X; wN) we also have a cap product homomorphism H*(X; M) -\[X]--!Hn-* (X; M Z wN) where the tensor product M Z wN is given the left -module structure via g . (x y) := xg-1 gy (g 2 ss; x y 2 M wN) : With respect to these conventions, there is a canonical isomorphism of left modules w ~= Z wZ. 2.1. Definition. The space X is called a Poincare complex of formal dimen- sion n if there is a class [X] 2 Hn(X; wZ) such that cap product with it induces an isomorphism ~ \[X]: H*(X; ) -=!Hn-* (X; w) : More generally, a disconnected space X is a Poincare complex of formal dimen- sion n if each of its connected components is. We abbreviate the terminology and refer to X as a Poincare n-complex. For the rest of the paper, we shall be implicitly assuming that X is connected. If the orientation character is trivial, we say that X is orientable, and a choice* * of fundamental class [X] in this case is called an orientation for X. 2.2. Remark. (1). Wall proved that the definition is equivalent to the assertion that the cap product map H*(X; M) -\[X]--!Hn-* (X; wM) is an isomorphism for all left -modules M. In particular, taking M = Z, we obtain the isomorphism \[X]: H*(X; Z) ~=Hn-* (X; wZ) as a special case, which amounts to the statement of the classical Poincare duality isomorphism when w is the trivial orientation character. (2). Every compact n-manifold X satisfies this form of Poincare duality.1 A vector bundle j over S1 is trivializable if and only if w1(j) = +1 2 H1(S1; Z2) = Z2 = {1}: The homomorphism w :ss -! {1} is defined by mapping a loop `: S1 -! X to +1 if the pullback of the tangent bundle of X along ` is trivializable and -1 otherwise. (3). In Wall's treatment of surgery theory [Wa4], the above definition of Poincare complex is extended to include simple homotopy information. This is done as follows: the cap product homomorphism is represented by a chain map C*(Xe; ) -! Cn-* (Xe; w) of finite degreewise free chain complexes of right -modules. One requires the Whitehead torsion of this chain map to be trivial. In this instance, one says that X is a simple Poincare n-complex. It is known that every compact manifold has the structure of a simple Poincare complex. ______________ 1The standard picture of a handle in a manifold, with its core and co-core i* *ntersecting in a point, has led Bruce Williams to the following one word proof of Poincare duali* *ty: BEHOLD! POINCARE DUALITY SPACES 5 2.3. Poincare pairs. Let (X; A) be a finite CW pair. Assume that X is connected. We assume that X comes equipped with a homomorphism w :ss1(X) -! {1}. We say that (X; A) is a Poincare n-pair if there is a class [X] 2 Hn(X; A; wZ) such that cap product with it induces an isomorphism ~= w \[X]: H*(X; ) -! Hn-* (X; A; ) : Moreover, it is required that @*([X]) 2 Hn-1 (A; wZ) equips A with the struc- ture of a Poincare complex, where the orientation character on A is the one induced by the orientation character on X. Note, however, that in many im- portant examples, A is not connected, even though X is. 2.4. Examples. We mention some ways of building Poincare complexes. Gluing. If (M; @M) and (N; @N) are n-manifolds with boundary or, more generally, Poincare pairs, and h: @M -! @N is a homotopy equivalence, then the amalgamated union M [h N is a Poincare n-complex. A special case of this is the connected sum X]Y of two Poincare complexes Xn and Y n. To define it, we need to cite a result of Wall: every Poincare n-complex X has the form K [ Dn , where K is a CW complex and dim K < n; this decomposition is unique up to homotopy (see 4.9 below). Converting the attaching map Sn-1 ! K into an inclusion Sn-1 K , we see that (K ; Sn-1 ) is a Poincare n-pair. Similarly, with Y = L [ Dn , we may define the connected sum X]Y to be K [Sn-1 L . Fibrations. Suppose that F ! E ! B is a fibration with F , E and B the the homotopy type of finite complexes. Quinn [Qu2] has asserted that E is a Poincare complex if and only if F and B are. A proof (using manifold techniques) can be found in a paper of Gottlieb [Got]. This result is important because it explains a wide class of the known ex- amples of Poincare complexes: (1) The total space of a spherical fibration over a manifold. (2) The quotient of a Poincare complex by a free action of a finite group. In a somewhat different direction, if a finite group G acts on a Poincare com- plex M, then the orbit space M=G satisfies Poincare duality with rational coefficients. This includes for example the case of orbifolds. S-duality. Let K and C be based spaces, and suppose that d: Sn-2 -! K ^ C is an S-duality map, meaning that slant product with the homology class d*([Sn-2 ]) 2 eHn-2(K ^ C) induces an isomorphism in all degrees f : eH*(K) ~= Hen-*-2 (C). Let P :(K ^ C) -! K _ C denote the generalized Whitehead prod- uct map, whose adjoint K ^ C -! (K _ C) is defined by taking the loop 6 JOHN R. KLEIN commutator [iK ; iC ] (Samelson product), where iK : K ! (K _ C) and iC :C ! (K _ C) are adjoint to the inclusions (see [B-S, p. 192]). The CW complex X := (K _ C) [POd Dn is a Poincare n-complex, with * \[X] = 0f f 0 : Hn-* (X) = eHn-*-1(K) eHn-*-1 (C) ~=H*(X) = eH*-1(K) eH*-1(C) (* 6= 0; n) : (The proof uses [loc. cit., 4.6, 5.14]; see also 4.10 below). Spaces of this k* *ind arise in higher dimensional knot theory, where X is the boundary of a tubular neighborhood of a Seifert surface V n Sn+1 (i.e., the double V [@V V of (V; @V )) of a knot Sn-1 Sn+1 . Given Xn as above, we can form a Poincare (n+ 2)-complex Y n+2 by ap- plying the same construction to the doubly suspended S-duality 2d: Sn ! K ^ C. Thus iterated application of the operation (K; C; d) 7! (K; C; 2d) gives rise to a periodic family of Poincare complexes. This type of phenomenon is related to the periodicity of the high-dimensional knot cobordism groups. 3. The Spivak fibration A compact smooth manifold Mn comes equipped with a tangent bundle oM , whose fibres are n-dimensional vector spaces. Embedding M in a high dimensional euclidean space Rn+k , we can define the stable normal bundle , which is characterized by the equation oM M = 0 in the reduced Grothendieck group of stable vector bundles over M. By iden- tifying a closed tubular neighborhood of Mn in Rn+k with the normal disk bundle D(), and collapsing its complement to a point (the Thom-Pontryagin construction), we obtain the normal invariant2 ff :Sn+k = (Rn+k )+ -collapse----!Rn+k =(Rn+k - intD(M )) -excision----!~T* * (M ); = in which T () = D(M )=S(M ) is the Thom space of (here, S(M ) denotes the normal sphere bundle of M ). The map ff satisfies U \ ff*([Sn+k ]) = [M] ; ______________ 2The use of this term in the literature tends to vary; here we have chosen t* *o follow Williams [Wi1]. POINCARE DUALITY SPACES 7 where U 2 Hk (D(M ); S(M ); Zt) denotes a Thom class for M , in which the latter cohomology group is taken with respect to the local coefficient system defined by the first Stiefel-Whitney class of M (i.e., the orientation charact* *er of M). The above relation between the normal invariant, the Thom class and the fundamental class is reflected in an observation made by Atiyah. If p: D(M ) ! M denotes the bundle projection, then the assignment v 7! (v; p(v)) defines a map of pairs (D(M ); S(M )) -! (D(M ) x M; S(M ) x M) which induces a map of associated quotients T (M ) -! T (M ) ^ M+ ; where M+ denotes M with the addition of a disjoint basepoint. Composing this map with the normal invariant, we obtain a map Sn+k d-!T (M ) ^ M+ : 3.1. Theorem. (Atiyah Duality [At]). The map d is a Spanier-Whitehead duality map, i.e., slant product with the class d*([Sn+k ]) 2 eHn+k(T (M ) ^ M+* * ) yields an isomorphism eH*(T (M )) ~= eHn+k-*(M+ ) : With respect to this isomorphism (or rather, taking a version of it with twisted coefficients), we see that a Thom class U maps to a fundamental class [M] and the map is given by cap product with ff*([Sn+k ]). Thus, the relation U \ ff*([Sn+k ]) = [M] is a manifestation of the statement that the Thom complex T (M ) is a Spanier-Whitehead dual of M+ . The above discussion was intended to motivate the following: 3.2. Definition. Let X be a Poincare n-complex with orientation character w. By a Spivak normal fibration for X, we mean o a (k- 1)-spherical fibration p: E -! X, and o a map Sn+k ff-!T (p) ; where T (p) = X [ CE denotes the mapping cone of p. Moreover, we require that U \ ff*([Sn+k ]) = [X] ; where U 2 Hk (p; Zw ) is a Thom class for the spherical fibration p (here we are taking the cohomology group of the pair (X [p E x I; E x 0) defined by the mapping cylinder of p and the coefficients are given by the local system on X defined by the orientation character w). The map ff :Sn+k -! T (p) is called a normal invariant. 8 JOHN R. KLEIN 3.3. Theorem. (Spivak). Every Poincare n-complex X admits a Spivak nor- mal fibration with fibre Sk-1 , provided that k n. Moreover, it is unique in the following sense: given two Spivak fibrations (E0; p0; ff0) and (E1; p1; ff* *1) with respect to the same integer k, then there exists a stable fibre homotopy equivalence h :E1 -'!E2 such that the induced map T (h) :T (p0) -! T (p1) composed with ff0 is homotopic to ff1. Actually, Spivak only proves this in the 1-connected case, but a little care shows how to extend to result to the non-simply connected case. Let me now give Spivak's construction. As X is a finite complex, we can identify it up to homotopy with a closed regular neighborhood N of a finite polyhedron in euclidean space Rn+k . Let p: E -! X be the result of converting the composite @N -! N ' X into a fibration. One now argues that the homotopy fibre of p is homotopy equivalent to a (k- 1)-sphere. To see this, we combine n-dimensional Poincare duality for X together with the (n+k)-dimensional Poincare duality for (N; @N) (the latter having trivial orientation character) to conclude that H*(X; ) ~= Hn-* (X; w) ~= Hn-* (N; w) ~= Hk+* (N; @N; (w )e) ~= Hk+* (p; (w )e) ; where (w )e denotes the effect of converting w to a right module by means of the trivial orientation character e(g) := 1. Now, it is straightforward to check that this isomorphism is induced by cup product with a class U 2 Hk (p; Zw ), so it follows that the fibration p: E -! X satisfies the Thom isomorphism with respect to twisted coefficients. However, by the following, such fibrations are spherical fibrations. 3.4. Lemma. (Spivak [Spi, 4.4], Browder [Br4, I.4.3]). Suppose that p :E ! B is a fibration of connected spaces whose fibre F is 1-connected. Then F ' Sk-1 , k 2, if and only if the generalized Thom isomorphism holds, i.e., there exists a class U 2 Hk (p; Zw ) (with respect to some choice of orientation character w :ss1(B) -! {1}) such that cup product induces an isomorphism U[ :H*(B; ) -! H*+k (p; (w )e) : (The original proof of this lemma involves an intricate argument with spectral sequences. For an alternative, non-computational proof see Klein [Kl1].) To complete the proof of the existence of the normal fibration, we need to construct a normal invariant ff :Sn+k -! T (p). By definition, T (p) is homoto* *py equivalent to N=@N, nd the latter comes equipped with a degree one map Sn+k -! N=@N POINCARE DUALITY SPACES 9 given by collapsing the exterior of N to a point. This defines ff. Observe that when X is a smooth manifold then the Spivak fibration E -! X admits a reduction to a k-plane bundle with structure group O(k), i.e., the stable normal bundle of X. Similar remarks apply to PL and topological manifolds. This observation gives the first order obstruction to a finding a closed (TOP, PL or DIFF) manifold which is homotopy equivalent to a given Poincare complex: the normal fibration should admit a (TOP, PL or DIFF) reduction. We wish to illustrate the utility of this by citing a result from surgery th* *eory. 3.5. Theorem. (Browder, cf. [Ra4, p. 210]). If X is a 1-connected Poincare complex of dimension 5, then X is homotopy equivalent to a closed topological manifold if and only if the normal fibration for X admits a TOP-reduction. As a corollary, we see that every finite 1-connected H-space of dimension 5 is homotopy equivalent to a topological manifold: the Spivak fibration in this case is trivializable (cf. Browder and Spanier [Br-Sp]), so we may take the trivial reduction. 3.6. An alternative approach. The above construction of the Spivak nor- mal fibration required us to identify the Poincare complex X with a regular neighborhood of a finite polyhedron in Rn. From an aesthetic point of view, it is desirable to have a construction which altogether avoids the theory of regul* *ar neighborhoods. The following, which was discovered by the author, achieves this. To simplify the exposition, we shall only consider the case when ss1(X) is trivial, and leave it to the reader to fill-in the details in the general case. Let G be a topological group (which to avoid pathology, we assume is a CW complex). Consider based G-spaces built up inductively from a point by attaching free G-cells Dj^G+ along their boundaries Sj-1 ^G+ . Such G-spaces are the free, based G-CW complexes. We shall call such G-spaces cofibrant. Given a cofibrant G-space Y , define the equivariant cohomology of Y by He*G(Y ) := eH*(Y=G; Z) where the groups on the right are given by taking reduced singular cohomology. Similarly, we have the equivariant homology of Y eHG*(Y ) := eH*(Y=G; Z) : Given two G-spaces Y and Z, we can form their smash product Y ^Z. Gives this the diagonal G-action, and let Y ^G Z denote the resulting orbit space. 3.7. Definition. Assume that ss0(G) is trivial. A map of based spaces d: Sm -! Y ^G Z is said to be an equivariant duality map if the correspondence x 7! x=d*([Sm ]) defines an isomorphism eH*G(Y ) -~=!eHGm-*(Z) : 10 JOHN R. KLEIN 3.8. Remarks. (1). Another way of saying this is that the evident composite Sm -! Y ^G Z -! (Y=G) ^ (Z=G) is an S-duality map. (2). Our definition is a dual variation of one given by Vogell [Vo], and the se* *t- up is similar to Ranicki [Ra1, x3] who defines an analogue for discrete groups. If G is not connected, then the definition is slightly more technical in that we have to take cohomology with = Z[ss0(G)]-coefficients. Now, using a cell-by-cell induction (basically, Spanier's exercises [Spa, pp. 462-463] made equivariant), one verifies that every finite cofibrant G-space Y (i.e., which is built up from a point by a finite number of G-cells) has the property that there exists a finite G-space Z and an equivariant duality map Sm -! Y ^G Z for some choice of m 0. It is well-known that any connected based CW complex X comes equipped with a homotopy equivalence BG -'!X, where G is a suitable topological group model for the loop space of X (e.g., take G to be the geometric realization of * *the underlying simplicial set of the Kan loop group of the total singular complex of X). Here, BG denotes the classifying space of X. Let EG be the total space of a universal bundle over X. Then EG is a free contractible G-space. Let EG+ be the effect of adjoining a basepoint to EG. Since BG is homotopy finite, it follows that EG+ is the equivariant type of a finite cofibrant G-space. Hence, there exists an equivariant duality map Sm d-!EG+ ^G Z := ZhG for suitably large m, where ZhG := (EG xG Z)=(EG xG *) is the reduced Borel construction of G acting on Z (note in fact that ZhG is homotopy equivalent to Z=G since Z is assumed to be cofibrant). In what follows, we assume that m n =: dim X. 3.9. Claim. If BG has the structure of an n-dimensional Poincare com- plex, then Z is unequivariantly homotopy equivalent to a sphere of dimension m- n- 1. Proof. Combining Poincare duality with equivariant duality, we obtain an iso- morphism eHm-n+* (ZhG ) ~=Hen-* (BG+ ) ~=He*(BG+ ) : One checks that this isomorphism is induced by cap product with a suitable class U 2 Hem-n (ZhG ). Now observe that up to a suspension, ZhG is the mapping cone of the evident map EG xG Z -! BG and it follows that ZhG amounts to the Thom complex for this map converted into a fibration. It follows that the Thom isomorphism is satisfied, and we conclude by 3.4 above that its fibre Z has the homotopy type of an (m- n- 1)- sphere. To complete our alternative construction of the Spivak fibration, we need to specify a normal invariant ff. This is given by the duality map d: Sm -! ZhG . POINCARE DUALITY SPACES 11 4. The classification of Poincare complexes We outline the classification theory of Poincare complexes in two instances: (i) low dimensions, and (ii) the highly connected case. In (i), we shall see th* *at the main invariants are of Postnikov and tangential type, and ones derived from them. In (ii), the Hopf invariant is the main tool. 4.1. Dimension 2. Every orientable Poincare 2-complex is a homotopy equiv- alent to a closed surface (see Eckmann-Linnell [E-L] and Eckmann-M"uller [E- M]). Surprisingly, this is a somewhat recent result. 4.2. Dimension 3. Clearly, Poincare duality implies that a 1-connected Poincare 3-complex X is necessarily homotopy equivalent to S3. Wall [Wa3] studied Poincare 3-complexes X in terms of the fundamental group ss = ss1(X), the number of ends e of ss and the second homotopy group G = ss2(X). The condition that e = 0 is the same as requiring ss to be finite. It follows that the universal cover of X is homotopy equivalent to S3, so G is trivial in this instance. It turns out in this case that ss is a group period 4, meaning that Z admits* * a periodic projective resolution of Z[ss] modules of period length 4. Wall showed that the first k-invariant of X is a generator g of H4(ss; Z) (the latter which* * is a group of order |ss|). The assignment X 7! (ss1(X); g) was proved to induce a bijection between the set of homotopy types of Poincare complexes and the the set of pairs (ss; g) with ss finite of period 4 and g 2 H4(ss; Z) a generat* *or, modulo the equivalence relation given by identifying (ss; g) with (ss0; g0) if * *there exists an isomorphism ss -! ss0 whose induced map on cohomology maps g0 to g. In the case when e 6= 0, then ss is infinite and Xe is non-compact. If e = 1, homological algebra shows that eXis contractible in this case, so X is a K(ss; * *1). If e = 2, the Wall shows that X is homotopy equivalent to one of RP3]RP3, S1 x RP2 or the one of the two possible S2-bundles over S1. This summarizes the classification results of Wall for groups for ss in which e 2. In 1977, Hendriks [He] showed that the homotopy type of a connected Poincare 3-complex X is completely determined by three invariants: o the fundamental group ss = ss1(X), o the orientation character w 2 Hom (ss; Z=2), and o the element o := u*([X]) 2 H3(Bss; wZ) given by taking the image of the fundamental class with respect to the homomorphism H3(X; wZ) -! H3(Bss1(X); wZ) induced by the classifying map u : X -! Bss for the universal cover of X. Call such data a Hendriks triple. Shortly thereafter, Turaev [Tu] characterized those Hendriks triples (ss; w;* * o ) which are realized by Poincare complexes, thereby completing the classification. For a ring , let ho-mod be the category of fractions associated to the cat- egory of right -modules given by formally inverting the class of morphisms 0 ! P , where P varies over the finitely generated projective modules. Call a homomorphism M ! N of right -modules a P -isomorphism if it induces an isomorphism in ho-mod . 12 JOHN R. KLEIN Set = Z[ss], where ss is a finitely presented group which comes equipped with an orientation character w :ss -! {1}. Let I denote the augmenta- tion ideal, given by taking the kernel of the ring map -! Z defined on group elements by g 7! 1. In particular, I is right -module. Choose a free right -resolution . .-.d3!C2 -d2!C1 -! I -! 0 of I, with C1 and C2 finitely generated. Let C* := hom (Ci; ) denote the corresponding complex of dual (left) modules. Let J be the right module given by taking the cokernel of the map *)w (C*1)w -(d2--!(C*2)w : Then Turaev shows that there is an isomorphism of abelian groups ~= w A: hom ho-mod (J; I) -! H3(ss; Z) : 4.3. Theorem. (Turaev). A Hendriks triple x := (ss; w; o ) is realized by a Poincare 3-complex if in only if o = A(t) for some P -isomorphism t :J ! I. 4.4. Dimension 4. Milnor [Mi2] proved that the intersection form H2(X4) H2(X4) -! Z (or equivalently, the cup product pairing on 2-dimensional cohomology) de- termines the homotopy type of a simply connected Poincare 4-complex, and that every unimodular symmetric bilinear form over Z is realizable. We should perhaps also mention here the much deeper theorem of Freedman, which says that the homeomorphism type of a closed topological 4-manifold is determined by its intersection form and its Kirby-Siebenmann invariant (the latter is a Z=2-valued obstruction to triangulation). We may therefore move on to the non-simply connected case. It is well- known that any group is realizable as the fundamental group of a closed 4- manifold, and hence of a Poincare 4-complex. Given a Poincare 4-complex X with fundamental group ss, the obvious invariants which come to mind are G := ss2(X) and the intersection form on the universal cover, which can be rewritten as OE: G x G -! Z (since ss2(X) = H2(Xe)); the group ss acts via isometries on the latter. Wall [Wa3] studied oriented Poincare 4-complexes X4 whose fundamental group is a cyclic group of prime order p 6= 2. Wall showed under these assump- tions that the homotopy type of X is determined by G and the intersection form G x G -! Z. However, when ss is the group of order 2, this intersection form is too weak to detect the homotopy type of X (see [H-K, 4.5]). Hambleton and Kreck [H-K] extended Wall's work to the case when ss is a finite group with periodic cohomology of order 4. To a given oriented X4, they associate a 4-tuple (ss; G; OE; k) POINCARE DUALITY SPACES 13 where ss = ss1(M), G = ss2(M), OE: G x G -! Z denotes the intersection form and k 2 H3(ss; G) denotes the first Postnikov invariant of X. Such a sys- tem is called the quadratic 2-type of X. Moreover generally, one can consider all such 4-tuples, and define isometry (ss; G; OE; k) -! (ss0; G0; OE0; k0) co* *nsist of isomorphisms ss ~=ss0 and G ~=G0 which map OE to OE0 and k to k0. 4.5. Theorem. (Hambleton-Kreck). Let X4 be a closed oriented Poincare complex with ss = ss1(X) a finite group having 4-periodic cohomology. Then the homotopy type of X is detected by the isometry class of its quadratic 2-type. Notice that the result fails to identify the possible quadratic 2-types which occur for Poincare complexes. Bauer [Bauer] extended this to finite groups ss whose Sylow subgroups are 4-periodic. Teichner [Te] extended it to the non-orientable case where a certain additional secondary obstruction appears. Teichner also realizes the obstruction by exhibiting a non-orientable Poincare 4-complex having the same quadratic 2-type as RP4]CP2, but the two spaces have different homotopy types. Thus Teichner's secondary obstruction may be non-trivial. Other examples in the non-orientable case were constructed by Ho, Kojima, and Raymond [H-K-R]. Another approach to classification in dimension 4 is to be found in the works of Hillman (see e.g., [Hill]). We should also mention here the work of Baues [Baues] which a provides a (rather unwieldy but) complete set of algebraic invariants for all 4-dimensional CW complexes. 4.6. Dimension 5. The main results in this dimension assume that the fun- damental group is trivial. Madsen and Milgram [M-M, 2.8] determined all Poincare 5-complexes with 4-skeleton homotopy equivalent to S2 _ S3. They show that such a space is homotopy equivalent to one of the following: (1) S2 x S3, (2) S(j ffl2) = the total space of the spherical fibration that is given by taking the fibrewise join of the Hopf fibration S3 -j!S2 with the trivi* *al fibration ffl2: S2 x S1 -! S2, or (3) the space given by attaching a 5-cell to S2 _ S3 by means of the map S4 -! S2_S3 given by [2; 3]+j22, where [2; 3]: S4 ! S2_S3 denotes the attaching map for the top cell of the cartesian product S2xS3 (= the Whitehead product), j2 :S4 -! S2 denotes the composite j :S4 -! S3 followed by j, and 2: S2 ! S2 _ S3 denotes the inclusion. The last of these cases is the Gitler-Stasheff example mentioned in the introdu* *c- tion, and hence fails to have the homotopy type of a closed smooth 5-manifold. This can be seen by showing that the Thom space of the associated Spivak normal bundle fails to be the Thom space of a smooth vector bundle. St"ocker has completely classified 1-connected Poincare 5-complexes up to oriented homotopy type. To a given oriented X5, we may associate the system of invariants I(X) := (G; b; w2; e) where o G = H2(X), 14 JOHN R. KLEIN o b: T (G) x T (G) -! Q=Z is the linking form for the torsion subgroup T (G) G, o w2 2 Hom (G; Z=2) is the second Stiefel-Whitney class for the Spivak fibration of X (which makes sense since Hom (ss2(BSG); Z=2) = Z=2, where the space BSG classifies oriented stable spherical fibrations), a* *nd o e 2 H3(X; Z=2) ~= G Z=2 denotes the obstruction linearizing the Spivak-fibration over the 3-skeleton of X (we are using here that the map BSO ! BSG is 2-connected, so a linearization always exists over the 2-skeleton). We remark that the first three of these invariants was used by Barden [Bar] to classify 1-connected smooth 5-manifolds. More generally, one can consider tuples (G; b; w2; e) in which G is a finite* *ly generated abelian group, b: T (G) x T (G) -! Q=Z is a nonsingular skew sym- metric form, w2: G -! Z=2 is a homomorphism and e 2 G Z=2 is an ele- ment. The data are required to satisfy w2(x) = b(x; x) for all x 2 T (G) and (w2 id)(e) = 0. It is straightforward to define isomorphism and direct sums of these data, so we may define J to be the semi-group of isomorphism classes of such tuples. 4.7. Theorem. (St"ocker [Sto]). The assignment X5 7! I(X5) defines an isomorphism between J and the semigroup of oriented homotopy types of 1- connected Poincare 5-complexes, where addition in the latter is defined by con- nected sum. Using a slightly different version of this, it is possible to write down a c* *om- plete list of oriented homotopy types of 1-connected Poincare 5-complexes in terms of `atomic' ones and the connected sum operation (see [loc. cit., 10.1]). 4.8. The highly connected case. In "Poincare Complexes: I", Wall an- nounces that the classification of `highly connected' Poincare complexes will appear in the forthcoming part II. Unfortunately, part II never did appear. We shall recall some of the homotopy theory which would presumably enter into a hypothetical classification in the metastable range. To begin with, it is well-known that a closed n-manifold can be given the structure of a finite n-dimensional CW complex with one n-cell. The analogue of this for Poincare complexes was proved by Wall [Wa3, 2.4], [Wa4, 2.9] and is called the disk theorem: 4.9. Theorem. (Wall). Let X be a finite Poincare n-complex. Then X is homotopy equivalent to a CW complex of the form L [ffDn . If n 6= 3 then L can be chosen as a complex with dim L n- 1 (when n = 3, L can be chosen as finitely dominated by a 2-complex). Moreover, the pair (L; ff) is unique up to homotopy and orientation. Suppose that X is a n-dimensional CW complex of the form (K) [ffDn , with K connected. We want to determine which attaching maps ff :Sn-1 -! K give X the structure of a Poincare complex. To this end, we recall the James-Hopf invariant ssn-1 (K) -H! ssn-1 (K ^ K) POINCARE DUALITY SPACES 15 which is defined using the using the well-known homotopy equivalence J(K) -'! K, where J(K) denotes the free monoid on the points of K. In terms of this identification,QH is inducedQby the map J(K) ! J(K ^ K) given by mapping a word ixi to the word i (-1)rflij ifj < i; : H(fi n-1 i) 2 ssn-1 (S ) = Z ifi = j : Therefore, the obstruction to X satisfying Poincare duality is given by the demanding that matrix (ei[ ej) be invertible. For the classification (of manifolds) in the odd dimensional case n = 2r + 1, see [Wa3]. 5. Poincare embeddings The notion of Poincare embedding is a homotopy-theoretic impersonation of what one obtains from an embedding of actual manifolds. If a manifold X is decomposed as a union X = K [A C where K; C X are codimension zero submanifolds with common boundary A := K \ C, then X stratifies into two pieces, with A as the codimension one stratum and int(K q C) as the codimension zero stratum. By replacing the above amalgamation with its homotopy invariant analogue, i.e., the homotopy colimit of K- A -! C, we may recover X up to homotopy equivalence. A Poincare embedding amounts to essentially these data, except that we do not decree the spaces to be smooth manifolds: the manifold condition is weakened to the constraint that Poincare duality is satisfied. Specifically, suppose that we are given a connected based finite CW complex Kk of dimension k, a Poincare n-complex Xn and a map f :K -! X. The definition of Poincare embedding which we give is essentially due to Levitt [Le1]. 5.1. Definition. We say that f Poincare embeds if there exists a commutative diagram of based spaces A ----! C ? ? i?y ?y K ----! X f such that o the diagram is a homotopy pushout, i.e., the evident map from the double mapping cylinder K x 0 [ A x [0; 1] [ C x 1 to X is a homotopy equivalence, and POINCARE DUALITY SPACES 17 o the image of [X] under Hn(X; wZ) -! Hn(i; f* wZ) induced by the boundary map in Mayer-Vietoris sequence of the diagram gives (K ; A) the structure of an n-dimensional Poincare pair, where K := K [Ax0 A x [0; 1] denotes the mapping cylinder of i. Similary, [X] makes (C ; * *A) into a Poincare pair. o The map i is (n- k- 1)-connected. The space C is called the complement. The above definition applies when X has no boundary. If (X; @X) is a Poincare n-pair, then the definition is analogous, except that we require the map @X -! X to factor as @X -! C -! X. The first condition of the definition says that X is homotopy theoretically a union of K with its complement. The second condition says that the `strat- ification' of X is `Poincare'. The last condition is essentially technical. In * *the smooth category, it would be an automatic consequence of transversality (a closed regular neighborhood N a k-dimensional subcomplex of an n-manifold has the property that @N N is (n- k- 1)-connected), so the condition that i be (n- k- 1)-connected is imposed to repair the lack of transversality in the Poincare case. However, note when k n- 3 that i is 2-connected if and only if i is (n- k- 1)-connected, by duality and the relative Hurewicz theorem. We will assume throughout that we are in codimension 3, i.e., k n- 3. 5.2. Remark. Suppose additionally that Kk has the structure of a Poincare k-complex. Then application of 3.4 above shows that the homotopy fibre of i is homotopy equivalent to an (n- k- 1)-sphere. Hence the map i in the definition may be replaced by a spherical fibration. This recovers the notion of Poincare embedding given by Wall [Wa2, p. 113]. The following result, which has a `folk' co-authorship, says that the descent problem for finding locally flat PL-manifold embeddings can always be solved in codimension 3. Moreover, the smooth version can always be solved in the metastable range. 5.3. Theorem. (Browder-Casson-Sullivan-Wall [Wa2, 11.3.1]). (1). Suppose that Kk and Xn are PL manifolds and that k n- 3. Then f is homotopic to a locally flat PL embedding if and only if f Poincare embeds. (2). If Kk ; Xn are smooth manifolds with k n- 3, then f :Kk -! Xn is homotopic to a smooth embedding if and only if f Poincare embeds and, addi- tionally, one of the following holds: (i) 2n 3(k+ 1), or (ii) f is homotopic to a smooth immersion. Thus, the problem of finding an embedding of PL-manifolds in codimen- sion 3 has been reduced to a problem in homotopy theory. When can this homotopy problem be solved? A map Mm ! Nn of manifolds with n 2m+ 1 is always homotopic to an embedding, by transversality. It is natural to ask whether a similar result hol* *ds in the Poincare case. Fix a map f :Kk -! Xn , where Kk is a k-dimensional 18 JOHN R. KLEIN CW complex, Xn is a Poincare complex (possibly with boundary) and k n- 3. According to Levitt [Le1], f Poincare embeds when n 2k + 2 . One would expect that the result holds in one codimension less, in analogy with manifolds, but this isn't known in general. However, Hodgson [Ho1] asserts that f will Poincare embed when n 2k + 1, with the additional assumptions that K is a Poincare complex and X is 1-connected. Both Hodgson and Levitt used manifold engulfing techniques to arrive at these results. Recently, the author [Kl2] proved a general result about Poincare embed- dings which implies the Levitt and Hodgson theorems as special cases: 5.4. Theorem. Let f :Kk -! Xn be an r-connected map with k n- 3. Then f Poincare embeds whenever r 2k - n + 2 : Moreover, the Poincare embedding is `unique up to isotopy' if strict inequality holds. (Two Poincare embedding diagrams for f are called isotopic if they are isomorphic in the homotopy category of such diagrams.) In contrast with the engulfing methods of Levitt and Hodgson, the author proves this result using purely homotopy theoretic techniques (a main ingredi- ent of the proof is the Blakers-Massey theorem for cubical diagrams of spaces, as to be found in [Good]). An old question about Poincare complexes is whether or not the diagonal X ! X x X Poincare embeds. As an application of the above, we have 5.5. Corollary. Let Xn be a 2-connected Poincare n-complex. Then the di- agonal X ! X x X Poincare embeds. Moreover, any two Poincare embeddings of the diagonal are isotopic whenever X is 3-connected. It would be interesting to know whether or not the corollary holds with- out the connectivity hypothesis. Clearly, the diagonal of a manifold Poincare embeds, by the tubular neighborhood theorem, so the existence of a diago- nal Poincare embedding for a Poincare complex is an obstruction to finding a smoothing. 5.6. Example. Let X be a finite H-space with multiplication : X xX ! X. Write X = X0 [ Dn using the disk theorem, and let ff :Dn -! X be the characteristic map for the top cell of X. Consider the commutative diagram X x Sn-1 -idxff---!X x X0 ? ? \?y ?ys X x Dn ---d-! X x X where the map s is given by (x; y) 7! (x; (x; y)), and the map d is given by (x; v) 7! (x; xff(v)). Then the diagram is a homotopy pushout and, moreover, the restriction of d to X x * X x Dn coincides with the diagonal. Hence, the diagram amounts to a Poincare embedding of the diagonal. POINCARE DUALITY SPACES 19 5.7. Poincare embeddings and unstable normal invariants. Another type of question which naturally arises concerns the relationship between the Spivak normal fibration and Poincare embeddings in the sphere. Suppose that Kk is a Poincare complex equipped with a choice of spherical fibration p: S(p) -! K with fibre Sj-1 . One can ask whether Kk Poincare embeds in the sphere Sk+j with normal data p. That is, when does there exist a space W and an inclusion S(p) W such that K [S(p)W is homotopy equivalent to Sk+j ? Obviously, if p isn't a Spivak fibration then there aren't any such Poincare embeddings. So the first obstruction is given by the existence of a normal invariant Sk+j -! T (p). More generally, let Kk be a k-dimensional CW complex which is equipped with a map g :A -! K. Let K be the mapping cylinder of g and assume that (K ; A) is an oriented Poincare n-pair. We want to know when there exists a Poincare embedding of K in Sn with normal data A -! K, i.e., when does there exist an inclusion of spaces A W such that K [A W is homotopy equivalent to Sn ? This problem specializes to the previous one by taking g to be a spherical fibration. Now, if the problem could be solved, then a choice of homotopy equivalence Sn -'! K [A W gives rise to a `collapse' map Sn -'! K [A W ' K [A W -! K [A * = T (g) where T (g) denotes the mapping cone of g :A -! K. By correctly choosing our orientation for (K ; A), we may assume that this map is of degree one. This prompts the following more general notion of normal invariant. 5.8. Definition. Given g :A -! K as above together with an orientation for (K ; A), we call the homotopy class of any degree one map Sn -! T (g) a normal invariant. The following result says that there is a bijective correspondence between normal invariants and isotopy classes of Poincare embeddings in the sphere with given normal data in the metastable range. It was first proven by Williams [Wi1], using manifold methods. A homotopy theoretic proof has been recently given by Richter [Ri1]. 5.9. Theorem. Suppose that 3(k + 1) 2n and n 6. Then Kk Poincare embeds in Sn with normal data g :A -! K if and only if there exists a normal invariant Sn -! T (g). Moreover, any two such Poincare embeddings of K which induce the same normal invariants are isotopic provided that 3(k + 1) < 2n. Richter [Ri2] has found some interesting applications of this result. For example, he has shown how it implies that the isotopy class of a knot Sn Sn+2 is determined by its complement X, whenever ss*(X) = ss*(S1) for * 1=3(n + 2); this extends a theorem of Farber by one dimension. 6. Poincare Surgery Controversy seems to be one of the highlights of this subject, so to avoid potential crossfire I'll begin this section with a quote from Chris Stark's mat* *h- 20 JOHN R. KLEIN ematical review [Stk] of the book Geometry on Poincare spaces, by Hausmann and Vogel [H-V]: The considerable body of work on these matters is usually referred to as "Poincare surgery" although other fundamental issues such as transversality are involved. These efforts involve several points of view and a number of mathematicians_the authors of the present notes identify three main streams of prior scholarship in their intro- duction and include a useful bibliography. Because of technical dif- ficulties and unfinished research programs, Poincare surgery has not become the useful tool proponents of the subject once hoped to deliver. For the sake of simplicity, I shall only discuss the results found in [H-V], which is now the standard reference for Poincare surgery. We begin by ex- plaining the fundamental problem of Poincare surgery. To keep the exposition simple, we only consider the oriented case. 6.1. Surgery. Quinn [Qu2] defines a normal space to be a CW complex X equipped with an (oriented) (k- 1)-spherical fibration pX : E -! X and a degree one map ffX : Sn+k -! T (pX ), where T (pX ) denotes the mapping cone = Thom space of pX (here the integer k is allowed to vary). We define the formal dimension of X to be n. Similarly, we have the notion of normal pair (X; A). A normal map of normal spaces from X to Y consists of a map f :X -! Y and an oriented fibre equivalence of fibrations b: pX -'! pY covering f such that the composite Sn+k ffX--!T (pX ) -T(b)--!T (pY ) coincides with ffY . Note that the mapping cylinder of f has the structure of a normal pair whose boundary is X q Y . Similarly, there is an evident notion of normal cobordism for normal maps. The obvious example of a normal space is given by a Poincare complex equipped with Spivak fibration. The central problem of Poincare surgery is to decide when a given normal map f :X -! Y of Poincare complexes is normally cobordant to a homotopy equivalence. Analogously, in the language of normal pairs, one wants to know when a normal pair (X; A), with A Poincare, is normally cobordant to a Poincare pair. The algebraic theory of surgery of Ranicki [Ra1-2], [Ra3] associates to a normal map of Poincare complexes f :X -! Y a surgery obstruction oe(f) 2 Ln(ss1(Y )) which coincides with the classical one if the given normal map comes from a manifold surgery problem. The principal result of Poincare surgery says that this is the only obstruction to finding such a normal cobordism, i.e., that the manifold and Poincare surgery obstructions are the same. According to Hausmann and Vogel, there are to date three basic approaches to Poincare surgery obstruction theory. The first is to use thickening theory to replace a Poincare complex with manifold with boundary, so that we can avail ourselves of manifold techniques, such as engulfing. This is the embodied in approach of several authors, in- cluding Levitt [Le2], Hodgson [Ho6] and Lannes-Latour-Morlet [L-L-M]. One POINCARE DUALITY SPACES 21 philosophical disadvantage of this approach is that, in the words of Browder, "a problem in homotopy theory should have a homotopy theoretical solution" [Qu1]. The second approach, undertaken by Jones [Jo1], also uses sophisticated manifold theory. The idea here is to equip Poincare complexes with the struc- ture of a patch space, which a space having an `atlas' of manifolds whose trans* *i- tion maps are homotopy equivalences, and having suitable transversality prop- erties. Lastly, we have the direct homotopy theoretic assault, which was first out- lined by Browder and which was undertaken by Quinn [Qu1-3]. If a map fi :Sj -! Xn is an element on which one wants to do surgery, then the homo- topy cofiber X [fiDj+1 has the homotopy type of an elementary cobordism, i.e., the trace of the would-be surgery. Moreover, as Quinn observes, if the surgery can be done then there is a cofibration sequence X0 -! X [fiDj+1 -! Sn-j where X0 is the `other end' of the cobordism. The composite map X X [fiDj+1 -! Sn-j is a geometric representative for a cohomology class which is Poincare dual to the homology class defined by fi. Quinn's idea [Qu3] is to find homotopy theoretic criteria (involving Poincare duality) to decide when a map X [fiDj+1 -! Sn-j extends to the left as a cofibration sequence, thus yielding X0. Hausmann and Vogel point out that these three approaches are imbued with a great deal of technical difficulty and none of them were completely overcome. We pigeonhole the book of Hausmann and Vogel by placing it within the first of these schools. 6.2. Poincare bordism. Under this title belong the fundamental exact se- quences of Poincare bordism found by Levitt [Le2], Jones [Jo1] and Quinn [Qu2]. Given a normal space X, we can let Pn(X) denote the bordism group of normal maps (f; b): Y -! X with X a normal space of formal dimension n and Y a Poincare n-complex, and where cobordisms are understood in the Poincare sense. Similarly, we can define Nn(X) to be the bordism group of normal maps (f; b): Y -! X. Then there is an exact sequence . .-.!Ln(ss1(X)) -! Pn(X) -incl-!Nn(X) -! Ln-1 (ss1(X)) -! . . . and moreover, an isomorphism Nn(X) ~= Hn(X; MSG), where the latter de- notes the homology of X with coefficients in the Thom spectrum MSG whose n-th space is the Thom space of the oriented spherical fibration with fibre Sn-1 over the classifying space BSGn. 6.3. Transversality. Let A be a finite CW complex and suppose that (D; S) is a connected CW pair such that A includes in D as a deformation retract. We also assume that the homotopy fibre of S D is (k- 1)-spherical. Given an inclusion S C, let Y denote the union D [S C. Roughly, we are thinking of the Y as containing a `neighborhood thickening' D of A in such a way that the `link' S of A in Y is a spherical fibration (up to homotopy). 22 JOHN R. KLEIN Let X be a Poincare n-complex and let f :X -! Y be a map. We say that f is Poincare transverse to A when (f-1 (D); f-1 (S)) and (f-1 (C); f-1 (S)) have the structure of Poincare n-pairs, and moreover, we require that the homotopy fibre of the map f-1 (S) -! f-1 (D) is also (k- 1)-spherical. Hence, if f is Poincare transverse to A, we obtain a stratification of X as a union of f-1 (D) with f-1 (C) along a common Poincare boundary f-1 (S). Moreover, it follows from the definition that f-1 (A) has the structure of a Poincare (n- k)-complex, so we infer that the inclusion f-1 (A) X Poincare embeds (with normal data f-1 (S)). The main issue now is to decide when a map f :X -! Y can be `deformed' (bordant, h-cobordant) so that it becomes Poincare transverse to the given A. The philosophy is that although one can always deform a map in the smooth case to make it transverse, there are obstructions in the Wall L-groups for the Poincare case, and the vanishing of these obstructions are both necessary and sufficient for Poincare transversality up to bordism. The algebraic L-theory codimension k Poincare transversality obstructions for k = 1; 2 are discussed in Ranicki [Ra3, Chap. 7]. Supposing in what follows that k 3, Hausmann and Vogel provide a criterion for deciding when f can be made (oriented) Poincare bordant to a map which is transverse to A [H-V, 7.11]. They define an invariant t(f) 2 Ln-k (ss1(A)) whose vanishing is necessary and sufficient to finding the desired bordism. If in addition f is 2-connected, then t(f) is the complete obstruction to making f transverse to A up to homotopy equivalence (i.e., Poincare h-cobordism) [loc. cit., 7.23]). Assertions of this* * kind can be found in the papers of Levitt [Le2],[Le4],[Le5], Jones [Jo1], and Quinn [Qu2]. For a general formulation, see [H-V, 7.11, 7.14]. 6.4. Handle decompositions. Given a Poincare n-pair (Y; @Y ), and a Poincare embedding diagram Sk-1 x Sn-k-1 ----! C ? ? \ ?y ?y Sk-1 x Dn-k ----! @Y we can form the Poincare n-pair (Z; @Z) := (Y [ Dk x Dn-k ; C [ Dk x Sn-k-1 ) ; where Dk x Dn-k is attached to Y by means of the composite Sk-1 x Dn-k -! @Y Y and Dk x Sn-k-1 is attached to @Y by means of the map Sk-1 x Sn-k-1 -! C. Call this operation the effect of attaching a k-handle to (Y; @Y * *). Note that there is an evident map Y -! Z. A handle decomposition for a Poincare complex Xn consists of a sequence of spaces W-1 -! W0 -! . .-.!Wn POINCARE DUALITY SPACES 23 (with W-1 = ;) and a homotopy equivalence Wn -'! X. Moreover, each Wj is the underlying space of a Poincare n-pair with boundary @Wj in such a way that Wj is obtained from Wj-1 a a finite number of j-handle attachments. Handle decompositions are special cases of Jones' patch spaces [Jo1]. 6.5. Theorem. ( [H-V, 6.1]). If X is a Poincare n-complex with n 5, then X admits a handle decomposition. 6.6. Appendix: a quick update on the finite H-space problem When Browder posed his question: Does every finite H-space have the ho- motopy type of a closed smooth manifold?, it wasn't known that there exist 1-connected finite H-spaces which are not the homotopy type of compact Lie groups (except for products with S7 or quotients thereof; see Hilton-Roitberg [H-R] and Stasheff [Sta, p. 22] for examples). We remarked in x3 that every 1-connected finite H-space Xn has the homo- topy type of a closed topological n-manifold. Browder [Br5] has noted in fact that the manifold can be chosen as smooth and stably parallelizable if n isn't of the form 4k+ 2. Using Zabrodsky mixing [Z] and surgery methods, Pedersen [Pe] was able to extend Browder's theorem to show that certain classes of finite H-spaces (some with non-trivial fundamental group) have the homotopy type of smooth manifolds. Recall that spaces Y and Z are said to have the same genus if Y(p) ' Z(p) for all primes p, where Y(p) denotes the Sullivan localization of Y at p. Among other things, Pedersen proved that when a finite H-space X happens to be 1-connected and has the genus of a 1-connected Lie group, then X has the homotopy type of a smooth, parallelizable manifold. Weinberger [We] has settled the `local' version of the problem: if P denotes a finite set of primes, then a finite H-space is P -locally homotopy equivalent to a closed topological manifold. 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