MULTIPLE DISJUNCTION FOR SPACES OF POINCARÉ EMBEDDINGS

MULTIPLE DISJUNCTION FOR SPACES OF POINCARÉ EMBEDDINGS THOMAS G. GOODWILLIE AND JOHN R. KLEIN Abstract. Still at it. Contents 1. Introduction 1 2. Some Language 6 3. Getting the ambient space to be connected 6 4. Theorems 1 and 2 are equivalent when r 1. 8 5. Categories of spaces which factorize a fixed map 8 6. A categorical model for function spaces 9 7. Truncation 11 8. Gluing 14 9. Poincaré spaces 19 10. Poincaré embeddings 20 11. The Poincaré embedding space 21 12. Theorems 1 and 2 when r = 0 24 13. Theorem 1 when r=1 24 14. Theorem 2 when r = 2 26 15. Theorem 2 when r > 2 28 16. Appendix: some results about cubical diagrams 31 References 32 1. Introduction The purpose of this article to prove multi-relative disjunction statements for spaces of Poincaré embeddings. The results are a step in a program which seeks to establish similar statements about smooth embedding spaces. Date: September 6, 2004. Both authors are partially supported by the NSF. 1

2 THOMAS G. GOODWILLIE AND JOHN R. KLEIN The idea is that certain problems about spaces of smooth embeddings can be reduced to two kinds of problems, one involving concordance spaces and surgery theory, and the other involving a problem in homotopy theory. This paper handles the homotopy theoretic step of the program. The manifold steps will be dealt with in another paper [G-K]. Let P be a Poincaré duality space of (formal) dimension n with boundary P decomposed as P = 0 P 01 P 1 P, in which 0 P and 1 P are Poincaré spaces with common boundary 01 P = 0 P 1 P. Think of 0 P and 1 P as the outer and inner boundaries of P. We usually suppress 01 P from the notation and write P = 0 P 1 P. Call this structure a boundary decomposition of P. Let Q be another Poincaré space of dimension n equipped with a boundary decomposition, and let N be a Poincaré space of dimension n whose boundary decomposes as N = 0 P D 0 Q With the convention that 0 N := 0 P and 1 N = D 0 Q, we get a boundary decomposition of N. Fix now a Poincaré embedding of Q in N which extends the inclusion 0 Q N. Roughly, this means we are given a complement Poincaré space, denoted N Q, whose boundary (N Q) decomposes as 0 P D 1 Q, and we are given a weak homotopy equivalence Q 1 Q (N Q) N, whose restriction to N is the inclusion. If (P, N) denotes the space of Poincaré embeddings of P in N which are fixed on 0 P, then one has a map (P, N Q) (P, N) (see 11 for the definition of these spaces). The simplest kind of disjunction question is to ask for the connectivity of this map, since this number is an upper bounded for the number of parameters one can take in a parametrized family of embeddings of P in N so that the family can be moved off of Q.

MULTIPLE DISJUNCTION 3 Fig. 1. In categories of manifolds, the usual transversality theorems enable one to move a k-parameter family of embeddings of P in N away from Q whenever the sum of k and the dimensions of the co-cores of all handles appearing in handle decompositions of both P and Q are sufficiently large. In the Poincaré duality category there are no such transversality results. Nevertheless, it turns out that one can estimate the connectivity of the above map, but the method of doing so is completely different. Definition 1.1. The homotopy codimension of P is n p if and only if: the pair (P, 0 P ) has relative cell dimension p up to homotopy (in the sense that (P, 0 P ) is the retract up to homotopy of a relative cell complex having dimension p); the map 1 P P is (n p 1)-connected. In this instance write hocodim P n p. Remarks. (1). If P is a smooth manifold having a handle decomposition relative to 0 P whose handles have index p, then hocodim P n p. (2). If n p 3, and (P, 0 P ) has relative cell dimension p up to homotopy, then hocodim P n p if and only if 1 P P induces an isomorphism of fundamental groups. (The if part follows by Poincaré duality with twisted coefficients and the relative Hurewicz theorem.) Multi-relative disjunction. In this case we wish to move P off of a finite disjoint union of a Poincaré embedded subspaces Q i of N, with say, 1 i r. The assumption is that P can be moved off of any proper union of the Q i. The desired result when r = 3 is depicted in figure 2:

4 THOMAS G. GOODWILLIE AND JOHN R. KLEIN Fig. 2. We formulate the problem in terms of spaces of Poincaré embeddings. Suppose one is given n-dimensional Poincaré spaces P, and Q 1,..., Q r each equipped with boundary decompositions. For S R := {1,..., r}, set Q S = i S Q i. Then Q S inherits a boundary decomposition from the Q i. Let N n be another n-dimensional Poincaré space, and assume that N is expressed as 0 P D 0 Q R. (Our convention will be that 0 N = 0 P and 1 N = D 0 Q R.) Suppose we are given Poincaré embedding Q R N which is fixed on 0 Q R. Then for each S we have a Poincaré embedding with complement N Q S. Q S N, Definition 1.2. Let (P, N Q S ) be the space of Poincaré embeddings of P in N Q S which are fixed along 0 P. Then we have a commutative r-cube of spaces (P, N Q ). Assumptions/Notation. hocodim Q i n q i, hocodim P n p and p, q i n 3. Set r Σ := (n q i 2). i=1

MULTIPLE DISJUNCTION 5 With respect to these assumptions, we can now state our main conjecture: Conjecture A. The r-cube is (1 p+σ)-cartesian. (P, N Q ) We also have a companion conjecture which involves the map from embeddings to functions. Let F (P, N Q S ) denote the space of functions from P to N Q S which are fixed along 0 P. Then we have a map of r-cubes (P, N Q ) F (P, N Q ) which can be considered as an (r+1)-cube. Denote it by Conjecture B. The (r+1)-cube is (n 2p 1+Σ)-cartesian. F (P, N Q ). F (P, N Q ) These conjectures will not be proved here. Fortunately, to prove the smooth manifold versions of the conjectures, it is in fact sufficient to establish weaker results: Theorem 1. The r-cube is ( p+σ)-cartesian. (P, N Q ) This is only off by one dimension from Conjecture 1. However, the following statement is off by n p 1 dimensions from Conjecture 2: Theorem 2. The (r+1)-cube is ( p+σ)-cartesian. F (P, N Q ) Theorems 1 and 2 are the main results of this paper.

6 THOMAS G. GOODWILLIE AND JOHN R. KLEIN 2. Some Language We work in Top, the category of compactly generated topological spaces, together with its Quillen model structure. Recall that weak equivalences of Top are the weak homotopy equivalences, the fibrations are the Serre fibrations and the cofibrations are the retracts of those inclusion maps which are obtained from attaching cells (i.e., relative cellular inclusions) A map A B of spaces is k-connected if for all choices of basepoint in B, the homotopy fiber is an (k 1)-connected space (a nonempty space is always ( 1)-connected). A weak equivalence is an - connected map. A space is homotopy finite if has the homotopy type of a finite cell complex. It will be assumed that the reader is familiar with the language of homotopy limits and colimits [B-K]. By definition, a (commutative) r-cube of spaces is a functor X from the poset of subsets of R = {1,..., r} to spaces. By considering the restriction of X to the poset of subsets of a fixed subset S R, we get an S -cube which is called the face of X determined by S. An r-cube X is k-cartesian when the induced map X u holim S X S is k-connected. Similarly, X is k-cocartesian when the map hocolim S R X S X R is k-connected. A cube of spaces is -(co)cartesian if it is k-(co)cartesian for every integer k. A cube of spaces is strongly (co)cartesian if every face of X is -(co)cartesian. The realization C of a (small) category C is the geometric realization of its nerve. The various categories of spaces used this paper are large. There are several ways make sense of the realization of such categories. One method is to stay within a fixed Grothendieck universe. We will implicitly take this approach. A functor C D is r-connected when the induced map of realizations is. 3. Getting the ambient space to be connected The following will enable us to restrict ourselves to the case when the ambient space is connected. Lemma 3.1. Theorems 1 and 2 hold for all N if they hold for all connected N.

MULTIPLE DISJUNCTION 7 Proof. We give the argument for Theorem 1. The proof for Theorem 2 is similar. The idea will be that (P, N) breaks up into a disjoint union of certain products of Poincaré embedding spaces whose ambient spaces are connected components of N. The reader is referred to 11 for the actual definition of (P, N). Let N = N α and P = P β α S β T be the decompositions of N and P into connected components. For each V T, set P V = P β. β T Observe that a Poincaré embedding of P in N determines a function t: T S by taking the induced map of path components. For every u S, we get a Poincaré embedding of P t 1 (u) in N u which is fixed on 0 P t 1 (u). Hence, for the given function t, we can define t (P, N) = u S (P t 1 (u), N u ). Let E be the set of functions from T to S which arise from Poincaré embeddings of P in N that are fixed on 0 P. Then we get a decomposition (P, N) = t E t (P, N). We also have (P, N Q ) = t E t (P, N Q ), Where for each U R t (P, N Q U ) denotes the space of Poincaré embeddings from P to N Q U (fixed on 0 P ) such that the composite π 0 (P ) π 0 (N Q U ) π 0 (N) coincides with t. The point now is that each t (P, N Q ) is a product of cubes of Poincaré embedding spaces, with each ambient space connected. Since homotopy limits commute with products, the conclusion of Theorem 1 is valid for the diagram t (P, N Q ). In our situation, homotopy limits will also commute finite disjoint unions, since the indexing

8 THOMAS G. GOODWILLIE AND JOHN R. KLEIN category (the poset of subsets of R) is connected. We conclude that Theorem 1 holds for (P, N Q ) We will henceforth assume that N is connected. 4. Theorems 1 and 2 are equivalent when r 1. Proposition 4.1. Assume r 1 and let j be an integer. (1). If F (P, N Q ) is j-cartesian, then (P, N Q ) is (min(j, 1 p+σ))-cartesian. (2). If (P, N Q ) is j-cartesian, then F (P, N Q ) is (min(j, p+σ))-cartesian. By setting j = p + Σ, we immediately obtain Corollary 4.2. Theorems 1 and 2 are equivalent for r 1. Proof of 4.1. If X Y is any map of cubes, then [G1, Prop. 1.6] asserts X Y is k-cartesian if X is k-cartesian and Y is (k+1)- cartesian. X is k-cartesian if X Y is k-cartesian and Y is k-cartesian. We will apply this to the map of cubes (P, N Q ) F (P, N). For this we need to compute the degree to which F (P, N) is cartesian. Observe that N Q is strongly cocartesian. Using the variant of the Blakers-Massey theorem for strongly cocartesian cubes [G1, Th. 2.3], one sees that N Q is also (1+Σ)-cartesian. The diagram F (P, N Q ) is therefore (1 p+σ)-cartesian (since taking function spaces decreases the decreases connectivity by the relative cohomological dimension of the pair (P, 0 P )). Then statement (1) follows using k = min(j, 1 p+σ), and statement (2) follows using k = min(j, p+σ). 5. Categories of spaces which factorize a fixed map For a fixed map of spaces A X, let T(A X) be the category in which an object is a space Y equipped with a factorization A i Y p Y Y X,

MULTIPLE DISJUNCTION 9 We call i Y and p Y the structure maps of Y. When specifying an object, the structure maps are ordinarily suppressed. A morphism f : Y Z is a weak homotopy equivalence of spaces which preserves structure maps (fi Y = i Z and p Z f = p Y ). The category T (j,d) (A X). Suppose j and d are fixed integers. Let T (j,d) (A X) denote the full subcategory of T (j,d) (A X) whose objects Y satisfy the structure map Y X is (j+1)-connected; for all local coefficient bundles E on X, the cohomology groups H (Y, A; E) vanish in degrees > d. 6. A categorical model for function spaces Suppose that A 0 A is a cofibration. The identity then gives a restriction functor ρ: T(A ) T(A 0 ). For an object X T(A 0 ), let F(A, X) = X\ρ denote the right fiber of ρ over X (more usually called the under category or comma category of the pair (ρ, X).) An object of F(A, X) is a pair (Y, h) with Y T(A ) an object and h: X ρ(y ) a map. A morphism (Y, h) (Z, h ) is a map g : Y Z of T(A ) such that ρ(g)h = h. Remark 6.1. A more accurate notation would be F(A, X rel A 0 ). Lemma 6.2. The realization of F(A, X) is weak homotopy equivalent to the function space of maps A X which extend the fixed map A 0 X. Proof. To avoid notational clutter, we give the proof when A 0 =. The general case is similar. Let maps(a, X) be the space of maps from A to X. Let J denote π 0 (maps(a, X)) = the set of homotopy classes of maps from A to X. We first assert that J is in bijective correspondence with the path components of F(A, X). We argue as follows: for each α J, choose a representative f α : A X. Then φ: J π 0 (F(A, X)) can be defined by φ(α) = (X, id), where the structure map for X is f α. An inverse to

10 THOMAS G. GOODWILLIE AND JOHN R. KLEIN φ is given by mapping an object (Y, h) to [h 1 i Y ], where h 1 denotes a choice of homotopy inverse for h. For an object y := (Y, h) F(A, X), define F(A, X) y to be the component of F(A, X) that contains y. To complete the proof, it will be enough to exhibit a weak equivalence F(A, X) y maps(a, X) α, where φ(α) = [y], and the right side is the component of maps(a, X) that contains α. We will give such a weak equivalence by choosing y carefully within its component. If f α : A X, represents α, we take Y to be the space given by factorizing f α id: A X X, into a cofibration A X Y, followed by an acyclic fibration Y X. Define h: X Y to be the inclusion. Let H(Y rel A X) be the topological monoid of self weak equivalences of Y which restrict to the identity on A X. Then [W, prop. 2.2.5] gives a weak equivalence between the classifying space of H(Y rel A X) and the realization of F(A, X) y : BH(Y rel A X) F(A, X) y. We will identify BH(Y rel A X). There a fibration H(Y rel A X) H(Y rel X) maps(a, Y ) α, where maps(a, Y ) α is the component of the mapping space containing the inclusion A Y (this serves as the basepoint). The second map is given by restricting a weak equivalence to A. Since the first map is a homomorphism of topological monoids, the fibration is principal, i.e., it is induced by a map maps(a, Y ) α BH(Y rel A X), which is a weak equivalence, because H(Y rel X) is weakly contractible. Finally, use the weak equivalence Y X to identify maps(a, Y ) α with maps(a, X) α. Section spaces. A variant of the above gives a discrete categorical model for section spaces. Suppose now we are given a fixed cofibration A 0 A and a fixed map A X. Then we have a restriction functor ρ: T(A X) T(A 0 X). Let Y T(A 0 X) be an object and define sec (A,A0 )(Y X)

MULTIPLE DISJUNCTION 11 to be the right fiber of ρ at Y. Its objects are pairs (Z, h) with Z T(A X) an object and h: Y ρ(z) a map. A morphism (Z, h) (Z, h ) is a map g : Z Z such that ρ(g)h = h. The proof of the following is basically identical to the proof of 6.2 (we will omit the details). Lemma 6.3. Assume that Y X is a fibration. Then the realization of sec (A,A0 )(Y X) has the homotopy type of the space sec (A,A0 )(Y X) = the space of sections of Y X along A X which restrict to the fixed section A 0 Y. In other words, sec (A,A0 )(Y X) is the fiber of the fibration maps(a, Y ) maps(a 0, Y ) maps(a0,y ) maps(a, X) at the evident basepoint. Corollary 6.4. Assume in addition that Y X is (r+1)-connected; the cohomology of (A, A 0 ), taken with all local coefficient bundles, vanishes in degrees > s. Then the category sec (A,A0 )(Y X) is (r s)-connected. Proof. Use 6.3 to identify the realization of sec (A,A0 )(Y X) with the section space. The result follows by elementary obstruction theory applied to the section space. 7. Truncation Let π be a group. Fix a Z[π]-chain complex K ; a connected space Y with fundamental group π, and a Z[π]-chain map C (Y ) K. A truncation of these data consists of a connected space A and a map A Y, the latter which is isomorphism of fundamental groups, such that the composite map of Z[π]-chain complexes C (A) C (Y ) K is a quasi-isomorphism. More generally, fix in addition a map of spaces α: U Y. Then a truncation relative to U consists of a factorization U A Y of α such that

12 THOMAS G. GOODWILLIE AND JOHN R. KLEIN A Y is a π 1 -isomorphism; the composite C (A, U) C (Y, U) K is a quasi-isomorphism, where C (Y, U) denotes Z[π]-chain complex which computes the relative homology of α. The truncation category T(α; K ) is the full subcategory of T(U Y ) consisting of the relative truncations. Proposition 7.1. Assume j 3. Suppose that K is chain homotopy equivalent to a chain complex of free Z[π]-modules which vanishes in degrees > j. Furthermore, suppose that the chain map C (Y, U) K is n-connected. Then T(α; K ) is (n j 1)-connected. Proof. If n < j there is nothing to prove. Case (1): n = j. This is the assertion that T(α; K ) is non-empty. It is direct consequence of [Kl1, 4.1]. Case (2): n = j+1. We wish to show that T(α; K ) is connected. Let A and B be two truncations relative to U. We can arrange it so that the maps A Y and B Y are fibrations. We can also arrange it that the maps U A and U B are cofibrations. It will be sufficient to construct a morphism from A to B. Since C (Y, U) K is n-connected, the map C (A, U) C (Y, U) is (n 1)-connected. By the five lemma, the map C (A) C (Y ) is also (n 1)-connected. The relative Hurewicz theorem then implies that A Y is (n 1)-connected. Similarly, B Y is (n 1)-connected. The assumptions imply that the cohomology of (A, U), taken with all local coefficient modules, vanishes in degrees > n 1. Using this, obstruction theory shows that there is a map A B which factorizes U Y. By the Whitehead theorem, any such map is necessarily a weak equivalence (and hence is a morphism of T(U Y )) since the diagram is commutative. C (A, U) C (Y, U) C (B, U) K

MULTIPLE DISJUNCTION 13 Case (3): n > j+1. By case (2), T(α; K ) is connected. Let A T(α; K ) be an object; we can assume without loss in generality that A Y is a fibration and U A is a cofibration. Then T(α; K ) may be identified the classifying space of the topological monoid G T(U Y ) (A) of self-maps of the object A (see [W, Prop. 2.2.5]). The classifying space functor increases connectivity by 1. So it suffices to prove that G T(U Y ) (A) is (n j 2)-connected. We observe that there is an inclusion G T(U Y ) (A) sec (A,U) (A Y ). One argues as in case (2) that this inclusion is actually an equality. Consequently, we are reduced the problem of determining the connectivity of the section space. Using obstruction theory, the section space is (n j 2)-connected, since A Y is (n 1)-connected, and the cohomology of (U, A), taken with all local coefficient bundles, vanishes in degrees > j (cf. 6.4). Cocartesian replacement. Let X be an r-cube of spaces indexed by the subsets of R = {1,..., r}. Fix also a map of spaces U X. Definition 7.2. D(U; X ) is the full subcategory of T(U X ) consisting of objects A such that the r-cube given by replacing X by A is -cocartesian. We call D(U; X ) the cocartesian replacement category of X relative to U. Assumptions. For the r-cube X, we require that X is n-cocartesian; X S is connected for every S; for every inclusion T S, the homomorphism of fundamental groups π 1 (X T ) π 1 (X S ) is an isomorphism (denote the fundamental group by π); if K is the mapping cone of the map of Z[π]-chain complexes C (U) holim S C (X S ), then K has vanishing cohomology in degrees > j (with respect to all local coefficient bundles). Lemma 7.3. Assume in addition that j 3. (n j r)-connected. Then D(U; X ) is

14 THOMAS G. GOODWILLIE AND JOHN R. KLEIN Proof. D(U; X ) is just the truncation category T(U X ; K ). The fact that X is an n-cocartesian implies that the map C (X, U) K is (n r+1)-connected. By 7.1, T(U X ; K ) is (n j r)-connected. 8. Gluing Suppose that A X is a fixed map and A B is a cofibration. Then gluing in B gives a functor B : T(A X) T(B X B) defined by Y Y B (note: Y B is abbreviated notation for Y A B). We will need to consider several gluing functors simultaneously. For i R, let B B i be a cofibration. For S R, form B S = colim i S B i. Recall from 5, that for fixed integers j and d, we have a category T (j,d) (B S X B S ), Letting S vary, we obtain an r-cube of categories T (j,d) (B X B ), whose functors are given by gluing in the B i. Assumptions. X is connected; the map B B i is s i -connected (s i 2). Theorem 8.1. Assume in addition d 3. Then the r-cube is (2 + j d + T (j,d) (B, X B ) r (s i 1))-cartesian. i=1 The proof of 8.1 will take up the remainder of this section. For each S R, define a category as follows: an object consists of C (j,d) (B S X B S )

for each T S a choice of object MULTIPLE DISJUNCTION 15 Z T T (j,d) (B T X B T ) ; for each T V S, a choice of map Z T B V T Z V. In addition, the collection {Z T } must satisfy the following conditions: the maps Z T B V T Z V are compatible, i.e., the data describe an object of lim T S T(j,d) (B T X B T ) ; for each T, the map Z T lim V T Z V is a fibration; for each T, the map Z T X B T is a fibration. A morphism {Z T } {Z T } is specified by choosing, for each T, compatible maps Z T Z T of T(j,d) (B T X B T ). Lemma 8.2. The forgetful functor gives a weak equivalence C (j,d) (B S X B S ) T (j,d) (B S X B S ). Proof. Let D (j,d) (B S X B S ) be defined like C (j,d) (B S X B S ) but without the condition about maps being fibrations. The forgetful functor D (j,d) (B S X B S ) T (j,d) (B S X B S ) has an inverse up to natural transformation given by On the other hand, the inclusion Y {Y B T } T S. C (j,d) (B S X B S ) D (j,d) (B S X B S ) also admits an inverse up to natural transformation, using the fact that maps functorially factor into acyclic cofibrations followed by fibrations. Definition 8.3. ([G1, 1.13]) An r-cube X of spaces (or simplicial sets) is a fibration cube if for every S R, the map is a (Kan) fibration. X S lim T S X T

16 THOMAS G. GOODWILLIE AND JOHN R. KLEIN Lemma 8.4. ([G1, 1.15]). If X is a fibration cube, then the map lim S X S holim X S S is a weak equivalence. Consequently, X is k-cartesian if and only if is a k-connected map. X lim X S S Lemma 8.5. The cube of simplicial sets is a fibration cube. N.C (j,d) (B X B ) Proof. To BE FILLED IN. Proof of Theorem 8.1. By 8.2 and 8.5, it will be enough to prove that every fiber of the Kan fibration N.C (j,d) (B X B ) lim N.C (j,d) (B S X B S ) S is (1 + j d + r i=1 (s i 1))-connected. Let z lim S C (j,d) (B S X B S ), be any basepoint. Then z amounts to data consisting of for each S, an object Z S T(B S X B S ); for each inclusion S T, a morphism h S T : Z S B T S Z T such that the following conditions are satisfied: (1) the morphisms are compatible. (2) The map Z S X B S is a fibration. (3) The map Z S lim T S Z T is a fibration. Then the maps Z S X B S together with X form a punctured (r+1)-cube. Let X denote its limit, and form the associated (r+1)- cube X. Then X is -cartesian. A zero simplex in the fiber of the fibration N.C (j,d) (B X) lim S N.C(j,d) (B S X B S )

MULTIPLE DISJUNCTION 17 at z consists of choosing an object Y T (j,d) (B X) together with a map Y X such that the cube obtained by replacing X with Y is -cocartesian; this is the same as specifying an object of the cocartesian replacement category D(B ; X ). More generally, the fiber is precisely the nerve of D(B ; X ). We will compute the connectivity of D(B ; X ) using 7.3. The conditions on Z S imply that that the mapping cone of the Z[π]-chain map C (B ) holim C (X S ) S has vanishing cohomology in degrees > d with respect to all local coefficient bundles. By 7.3, D(B ; X ) will be (1 + j d + r i=1 (s i 1))- connected if the (r+1)-cube X is (2+j+ i s i)-cocartesian. We will prove the latter using dual Blakers-Massey theorem for cubes (cf. [G1, Th. 2.6]). Let us pause to recall its statement. Suppose that X is an r-cube of spaces. Then each non-empty subset S R determines in an evident way an (r S )-dimensional sub-cube of X which meets the terminal vertex X R. Assume this sub-cube is k S -cartesian, and assume that k T k S for S T. For each partition {T α } of R into non-empty subsets, form u({t α }) = r 1 + α k Tα. The dual Blakers-Massey theorem states that X is k-cocartesian, where k is the minimum of the function u taken over all such partitions. We remark that a partition of R into non-empty subsets is the same thing as a specifying a set of proper, pairwise disjoint faces of X, such that each face meets the terminal vertex X R, and the union of the faces in the set gives X with its initial vertex removed. In our situation, we have an (r+1)-cube X, with the property that each face meeting the terminal vertex is strongly cocartesian. Hence, one can compute the numbers k S using [G1, Th. 2.3]. For this, we need to know the connectivities of the maps out of the initial vertex of the given face. Observe that there are two kinds of faces meeting the terminal vertex of X. The first kind has initial vertex Z S, for some S R (this will determine a face of dimension r+1 S ). The maps emanating out of Z S are of two kinds: Z S Z S i and Z S X B S. By our assumptions, the first of these is s i -connected, and the second is (j+1)-connected. Then, according to [G1, Th. 2.3], the face determined

18 THOMAS G. GOODWILLIE AND JOHN R. KLEIN by Z S is ( S r + j + 1 + i/ S s i )-cartesian. The other kind of face has initial vertex X B S (this will be an (r S )- cube. The maps eminating out of it are X B S X B S i, an s i -connected map. Then [G1, Th. 2.3], shows that this face is (1 + S r + s i )-cartesian. i/ S When considered as functions of S, these expressions are monotone decreasing on inclusions (this uses s i 2). A partition of the set of faces meeting the terminal vertex of X is represented by a pair ({S α }, {T β }), where S α, T β R; the S α and T β give a partition of R; the S α determine partition of the set of faces of the first kind; the T β determine a partition of the set of faces of the second kind. This gives a partition of the set {1,..., r + 1} whose elements are S α and T β {r+1}. Conversely, any partition {U γ } of {1,..., r+1} is represented in the above format by taking the set of those U γ which don t contain r + 1 to be the S α, with the remaining U γ, after removing r + 1, as the T β. Suppose that {S α } has l-elements and {T β } has m-elements. Note that l, m 1. Then the above computation shows that the value of the function u on this partition is r + (l(1 r + j) + S α + s i ) + (m(1 r) + T β + s i ), α β i/ S α i/ T β where the summed expressions in this formula are indexed over complement of the intersection of the S α and the and the complement of the intersection of the T β respectively (note: if l 2, the intersection of the S α is empty, so the first summation is from 1 to r in this case). Since the S α and the T β form a partition of R, we see that r s i + s i. i/ S α i=1 i/ T β s i

MULTIPLE DISJUNCTION 19 Consequently, using this inequality and a bit of rewriting, we see that the value of u on this partition is at least r 2r + (l + m)(1 r) + lj + s i. Since l, m 1, this expression is at least 2 + j + r i=1 s i. To complete the proof, we must exhibit a partition which having this value. Consider the partition with two elements corresponding to the R-cube X B and the 1-cube Z R X B R. Then the value of u in this case is r r r + (1 r + s i ) + (j + 1) = 2 + j + s i. i=1 i=1 We record the following lemma for future use. Lemma 8.6. With respect to the given assumptions, the (r+1)-cube T (j,d) (B, X B ) is -cartesian. Proof. (Fill in). i=1 T(B, X B ) 9. Poincaré spaces Let (X, X) be a homotopy finite pair of spaces. X is said to be a Poincaré duality space of (formal) dimension d if there exists a local coefficient system L which is pointwise free abelian of rank one, and a (fundamental) class [X] H d (X, X; L) such that the cap product homomorphism [X]: H (X; M) H d (X, X; L M) is an isomorphism in all degrees for all local coefficient systems M on X; if [ X] H d 1 ( X; L X ) denotes the image of [X] under the boundary homomorphism, then [ X]: H ( X; M) H d 1 ( X, L X M) is an isomorphism for all local coefficient systems M on X. In particular, X is itself a (closed) Poincaré space of dimension d 1. Note that the five lemma also shows that the homomorphism is an isomorphism. [X]: H (X, X; M) H d (X; L M)

20 THOMAS G. GOODWILLIE AND JOHN R. KLEIN Lemma 9.1. If (L, [X] ) is another such choice, then there exists a isomorphism of local systems L L which transfers [X] to [X]. It is the unique isomorphism L L with this property. Proof. Since each path component of X satisfies Poincaré duality, it will suffice to consider the path connected case. Choose a basepoint for X and let Λ denote the integral group ring of the fundamental group. Then Poincaré duality with respect to [X] gives an isomorphism H d (X, X; Z) = H 0 (X; L) = L. Similarly, duality using [X] gives another isomorphism of the left hand side with L. Consequently, L and L are isomorphic. The displayed isomorphism is compatible with fundamental classes since Poincaré duality gives isomorphisms H d (X, X; L) = Z = H d (X, X; L ) that identifies the fundamental classes with the integer +1. If there is an another isomorphism from L L which transfers [X] to [X], we can invert it to get an automorphism φ of L which preserves [X]. We need to show that φ is the identity. The induced automorphism φ of H 0 (X; L) = L is just φ. But the above duality isomorphism L = H d (X, X; Z) shows that φ corresponds to the identity map of H d (X, X; Z). Consequently, φ is the identity. For additional foundations of the theory of Poincaré duality spaces, see [Wa2] or [Kl3]. 10. Poincaré embeddings Let N and P be Poincaré spaces, where P is equipped with a boundary decomposition P = 0 P 01 P 1 P. We will also assume that N is equipped with a boundary decomposition N = 0 P 01 P 1 N. That is, ( 0 N, 01 N) and ( 0 P, 01 P ) coincide. Definition 10.1. A Poincaré embedding of P in N (subject to the boundary constraints) is a pair in which (f, C),

MULTIPLE DISJUNCTION 21 f : P N is a map extending the inclusion 0 P N N (in particular, we obtain a map 1 P 01 P 1 N N). C T( 1 P 01 P 1 N N) is an object. The following axioms are to be satisfied: (1) the evident map P 1 P C N is a weak equivalence. (2) Let C be the mapping cylinder of the structure map 1 P 01 P 1 N C. Then C is a Poincaré space with C = 1 P 01 P 1 N. The map f : P N is called the underlying map. The object C is called the complement. Note that the weak equivalence P C N gives rise to a homomorphism H (N, N) = H (P C, N) H (P C, N C) = H (P, P ), where homology can be taken with respect to any local coefficient bundle on N. Using this homomorphism, we obtain an alternative characterization of axiom (2) when hocodim P 3 (this is [Kl1, Lem. 2.3] when 0 P = ; the proof in the general case is essentially the same). Lemma 10.2. Assume axiom (1) holds. If hocodim P 3, then axiom (2) is equivalent to the following: there exists a Poincaré duality structure (L, [N]) for N, such that if [P ] is the image of [N] under the homomorphism H n (N, N; L) H n (P, P ; L P ), then (L P, [P ]) is a Poincaré duality structure for P. 11. The Poincaré embedding space Definition 11.1. Let (P, N) be the category whose objects are pairs in which ((M, h), C),

22 THOMAS G. GOODWILLIE AND JOHN R. KLEIN (Function Data) (M, h) F(P, N) is an object. Here, F(P, N) is defined using the cofibration 0 P P (cf. 6). In particular, M comes equipped with a map f M : P M which is fixed on 0 P. We also assume that the composite N N h M is a cofibration. In particular, since h is a weak equivalence, M is a Poincaré space with M = N. (Complement Data) C T( 1 P 1 N M) is an object such that the pair (f M, C) defines a Poincaré embedding of P in M, subject to the given boundary constraints. A morphism ((M, h), C) ((M, h ), C ) is specified by a map and also a map where k : (M, h) (M, h ) F(P, N) k (C) C T( 1 P 1 N M ), k : T( 1 P 1 N M) T( 1 P 1 N M ) is the pushforward functor defined using composition with k. Fixing the underlying map. The model is simpler when the function P N is held fixed: Definition 11.2. Fix a map f : P N extending 0 P N, Let f (P, N) T( 1P 1 N N) be the full subcategory whose objects C are such that the pair (f, C) is a Poincaré embedding.

MULTIPLE DISJUNCTION 23 The map from embeddings to functions. There is a forgetful functor p: (P, N) F(P, N) ((M, h), C) (M, h). There is also an inclusion functor i: f (P, N) Epd (P, N) C ((N, id), C), Note that the composition pi is the constant functor to the basepoint (N, id) F(P, N). Definition 11.3. The Poincaré embedding space (P, N) is the realization of the category (P, N). The Poincaré embedding space with fixed underlying map f : P N is the realization of the category (P, N), and is denoted by Let f f (P, N). F (P, N) denote the realization of F(P, N), with basepoint (N, id). Proposition 11.4. The sequence is a fibration up to homotopy. f (P, N) Epd (P, N) F (P, N). Proof. Use Quillen s Theorem B [Q2]. (FILL IN DE- TAILS). Observation. If j = n p 2 and d = n then we have a full inclusion f (P, N) T(j,d) ( 1 P 1 N N). Assume that f : P N Q R is fixed along 0 P. Then cube of embedding categories f (P, N Q ) is well defined. Lemma 11.5. The map of r-cubes is -cartesian. f (P, N Q ) T (j,d) ( 1 P 1 (N Q ), N Q )

24 THOMAS G. GOODWILLIE AND JOHN R. KLEIN Proof. Using 10.2, it is straightforward to check that the map of r-cubes is cartesian. For each S, the map f (P, N Q S) T (j,d) ( 1 P 1 (N Q S ), N Q S ) is the inclusion of a set of connected components. Apply 16.1 below. 12. Theorems 1 and 2 when r = 0 Theorem 2 when r = 0 states that the map (P, N) F (P, N) is ( p)-connected. If p > 0, there is nothing to prove, since every map of spaces is ( 1)-connected. If p = 0, then, up to homotopy, P is obtained from 0 P by taking a disjoint union with a finite collection of n-disks. Explicitly, P = 0 P T, where (T, 1 P ) has the homotopy type of a finite disjoint union of copies of (D n, S n 1 ). Theorem 2 is then a consequence of induction and a theorem of Wall [Wa3, 2.9] which shows how to Poincaré embed disks. This gives Theorem 2 for r = 0. Theorem 1 when r = 0 is the statement that (P, N) is ( p)- connected. There are three cases: Case p > 1: The statement has no content. Case p = 1: This is the assertion that (P, N) is non-empty. If P and N are manifolds, use transversality. In the Poincaré space case, the result is a special instance of the main theorem of [Kl2]. Case p = 0: This is the assertion that (P, N) is connected. Argue as in the p = 1 case. When r = 1, we have 13. Theorem 1 when r=1 N = 0 P D 0 Q. By convention, 0 N = 0 P and 1 N = D 0 Q. We are also given a fixed Poincaré embedding which is fixed on 0 Q. Q N,

We wish to prove that the map MULTIPLE DISJUNCTION 25 (P, N Q) (P, N) is (n p q 2)-connected. By 11.4 it suffices to prove that the functor f (P, N Q) Epd(P, N) is (n p q 2)-connected for all maps f : P N Q compatible with the boundary conditions. Notation. Set A = 1 P D. Set j := n p 2 and d := n. Using 11.5, we have an -cartesian square f (P, N Q) f T (j,d) (A 1 Q, N Q) f (P, N) T (j,d) (A 0 Q, N). It will be enough to show that the right vertical map of this square is (n p 2)-connected. The right vertical map factors as a composite of the gluing functor followed by a restriction functor: T (j,d) (A 1 Q N Q) Q T (j,d) (A Q, N) ρ T (j,d) (A 0 Q, N). We will show that each map in the factorization is sufficiently connected. Lemma 13.1. The restriction functor is (n p 1)-connected. ρ: T (j,d) (A Q, N) T (j,d) (A 0 Q, N) Proof. By 6.3 and 8.6, each homotopy fiber of ρ is identified with a certain section space. By 6.4 each of these section spaces is (n p q 2)- connected. Lemma 13.2. The functor Q: T (j,d) (A 1 Q N Q) T (j,d) (A Q, N) is (n p q 2)-connected. Proof. Apply 8.1 in the r = 1 case.

26 THOMAS G. GOODWILLIE AND JOHN R. KLEIN In this case we have 14. Theorem 2 when r = 2 N = 0 P D 0 Q 12. By 11.4, it will suffice to prove that f (P, N Q 12) f (P, N Q 1) f (P, N Q 2) f (P, N) is ( p+σ)-cartesian, for any map f : P N Q 12 compatible with the boundary conditions. In what follows, we take j := n p 2 and d := n. Then by 11.5, the map of squares (P, N Q ) T (j,d) ( 1 P 1 (N Q ) N Q ), when regarded as a 3-cube, is -cartesian. Hence, it will be enough to show that the square T (j,d) ( 1 P 1 (N Q ) N Q ) is ( p+σ)-cartesian. Set A := 1 P D. Then the square may be subdivided into the four squares displayed below: T (j,d) (A 1 Q 12 N Q 12 ) Q 2 T (j,d) (A 1 Q 1 Q 2 N Q 1 ) ρ 2 T (j,d) (A 1 Q 1 0 Q 2 N Q 1 ) Q 1 T (j,d) (A Q 1 1 Q 2 N Q 2 ) (I) Q 2 Q 1 T (j,d) (A Q 12 N) (II) ρ 2 Q 1 T (j,d) (A Q 1 0 Q 2 N) ρ 1 (III) ρ 1 T (j,d) (A 0 Q 1 1 Q 2 N Q 2 ) Q2 T (j,d) (A 0 Q 1 Q 2 N) (IV) ρ 1 ρ 2 T (j,d) (A 0 Q 12 N). It will suffice to show that each square is ( p+σ)-cartesian. Lemma 14.1. The square (IV) is -cartesian. Proof. By 8.6 it is enough to prove that the version of square (IV) without the (j, d) decoration is -cartesian. Choose an object Y T(A 0 Q 1 Q 2 N). We can assume without loss in generality that Y N is a fibration.

MULTIPLE DISJUNCTION 27 Then by 6.3, the map of vertical homotopy fibers of square (IV) is, up to homotopy, the induced map of section spaces sec (Q12, 0 Q 1 Q 2 )(Y N) sec (Q1 0 Q 2, 0 Q 12 )(Y N). But this map is homeomorphism. Lemma 14.2. Squares (II) and (III) are (1 p + Σ)-cartesian. Proof. We will show that square (II) is (1 p + Σ)-cartesian. The argument for square (III) is gotten by switching the roles of Q 1 and Q 2. By [G1, prop. 1.18], it is enough to prove that the induced map of horizontal homotopy fibers is (1 p + Σ)-connected (for all choices of basepoints defining the homotopy fiber of the top horizontal map). To compute the map of homotopy fibers, choose an object Y T (j,d) (A 1 Q 1 0 Q 2 N Q 1 ) an object Z T (j,d) (A Q 1 0 Q 2 N), and a morphism Y Q 1 Z. Since maps can be functorially factored into acyclic cofibrations followed by fibrations, we may suppose without loss in generality that the map Y N Q 1 is a fibration, the map Z N is a fibration, and the restriction Y Z is a fibration. By 6.3, the homotopy fiber at Y of the top horizontal map of square (II) is identified with the section space sec (Q2, 0 Q 2 )(Y N Q 1 ), Similarly, the homotopy fiber of the bottom horizontal map of square (II) is identified with The map of homotopy fibers sec (Q12,Q 1 0 Q 2 )(Z N). sec (Q2, 0 Q 2 )(Y N Q 1 ) sec (Q12,Q 1 0 Q 2 )(Z N) arises from the -cocartesian square Y N Q 1 by taking spaces of sections associated with the vertical maps. Using the assumptions on j and d, the map Y Z is (n q 1 1)-connected, Z N

28 THOMAS G. GOODWILLIE AND JOHN R. KLEIN and the map Y N Q 1 is (n p 1)-connected. By the Blakers- Massey theorem, the square is (n p 1) + (n q 1 1) 1 = (2n p q 1 3)-cartesian. Since (Q 2, 0 Q 2 ) is the retract of a relative cell complex of dimension q 2, the connectivity of the map of section spaces is given by subtracting q 2 from the degree to which the square is homotopy cartesian. Consequently the map of section spaces is (2n p q 1 3) q 2 = (1 p + Σ)-connected. Now consider square (I). The maps in it are all gluing functors, so it is a special case of the cube of section 8, in which T (j,d) (B, X B ) B := A 1 Q 12, X := N Q 12, B i is obtained from B by amalgamating Q i, j := n p 2, d := n and s i = n q i 1. Then according to Theorem 8.1, the square T (j,d) (B X) B 2 T (j,d) (B 2 X B 2 ) B 1 T (j,d) (B 1 X B 1 ) B 1 B 2 T (j,d) (B 12 X B 12 ) is ( p+σ)-cartesian. In this instance we have 15. Theorem 2 when r > 2 N = 0 P D 0 Q R (with the convention that 0 N = 0 P and 1 N = D 0 Q R ). We wish to prove that the r-cube (P, N Q ) is (1 p+σ)-cartesian. The steps of the proof are the same as those in the r = 2 case.

MULTIPLE DISJUNCTION 29 Step 1. For any map f : P N Q R satisfying the boundary conditions, it will suffice to prove that the r-cube is ( p+σ)-cartesian. f (P, N Q ) Step 2. Set j := n p 2 and d := n. Then the map of r-cubes f (P, N Q ) T (j,d) ( 1 P 1 (N Q ), N Q ) is -cartesian, by 11.5. Consequently, it will be enough to show that T (j,d) ( 1 P 1 (N Q ), N Q ) is ( p+σ)-cartesian. Step 3. The latter cube can be subdivided into 2 r subcubes. The maps of the subdivided diagram are of the form: glue in Q i ( Q i ) or restrict from Q j to 0 Q j (ρ j ). The diagram can be described as follows: The 3 r vertices of it are indexed by the partitions of R into three subsets. A concise way of specifying such a partition is to give a function t: R {1, 2, 3}. Define a partial ordering on functions by if and only if s t t 1 (1) s 1 (1), s 1 (2) t 1 (2) and t 1 (3) s 1 (3) (note the change in variance in the second term). This partial ordering gives a poset with exactly 3 r elements. The shape of the poset is that of an r-cube which is midpoint subdivided into 2 r sub-cubes. For each function t, set Q t := 1 Q t 1 (1) 0 Q t 1 (2) Q t 1 (3). Setting A = 1 P D, we get a functor with the following properties: t T (j,d) (A Q t, N Q t 1 (1)) (1) the functors of the diagram are of the form Q i and ρ j. (2) The initial vertices of the 2 r sub-cubes are indexed by the functions t: R {1, 2, 3} satisfying t 1 (2) =. (3) The cube T (j,d) ( 1 P 1 (N Q ) N Q ) is indexed by those functions t satisfying t 1 (3) =.

30 THOMAS G. GOODWILLIE AND JOHN R. KLEIN Step 4. We will be done if we can show that each of the 2 r sub-cubes is ( p+σ)-cartesian. We group the sub-cubes into 3 types: Type 1. Cubes with initial vertex t in which t 1 (3) 2. We claim that such a cube is -cartesian. In this instance, there are at least two distinct restriction functors ρ i and ρ j in the cube. This means that the cube can be written as an (r 2)-cube of squares, with each square -cartesian (since all the maps within such a square are restriction functors and therefore 14.1 applies). The claim then follows from 16.2 below. Type 2. Cubes with initial vertex t in which t 1 (3) = 1. We claim that such a cube is (1 p+σ)-cartesian. This is analogous to the square (II) case of the previous section. The point is that if t 1 (3) = {j}, then such a cube can be written as a map C D of (r 1)-cubes, where each C S D S is the restriction functor ρ j and the maps in each of the cubes C and D are gluing functors. We index C and D using the subsets of R {j}. Choose compatible basepoints in D, and define F S = homotopy fiber(c S D S ). By [G1, prop. 1.18], it will suffice to show that the cube F is (1 p+σ)- cartesian. By 6.3, F S can be interpreted as a section space of a certain fibration Y S N Q R (S {j}), where sections are taken along Q j and fixed on 0 Q j. Moreover, the r-cube Y N Q R ( {j}) is strongly cocartesian. The maps out of the initial vertex Y are of two kinds. The first kind is of the form Y Y i, which is (n q i 1)-connected (using [Kl1, 5.6]). The other is the map Y N Q R {j}, which is (n p 1)-connected. Applying [G1, Th. 2.3], we infer that Y is 1 r+(n p 1)+ i j (n q i 1) = n p 1+ i j (n q i 2))-cartesian. Taking sections along Q j reduces connectivity by q j. So we get that the cube F is (1 p+σ)-cartesian.

MULTIPLE DISJUNCTION 31 Type 3. The cube with initial vertex t in which t 1 (3) =. We claim that this cube is ( p+σ)-cartesian. This is a cube of consisting entirely of the gluing functors Q i, for i R. By 8.1, this cube is ( p+σ)-cartesian (using s i := n q i 1). 16. Appendix: some results about cubical diagrams Lemma 16.1. Suppose X Y is a map of r-cubes such that each map X S Y S is the inclusion of a set of connected components. Then X Y is -cartesian if it strongly cartesian. Proof. The case r = 0 is trivial. If r = 1, then we wish to prove that the square X Y X 1 Y 1 is -cartesian assuming it is cartesian. For any basepoint x X 1, let F x be the homotopy fiber of X X 1. Then F x is homeomorphic to the homotopy fiber at x of Y Y 1 (using the fact that the diagram is cartesian and the horizontal maps are the inclusions of a union of connected components). Then the diagram is -cartesian (cf. [G1, prop. 1.18]). When r > 1, write X Y as an (r 2)-cube of cartesian squares. Each such square is then -cartesian by the r = 1 case. Now apply 16.2 below. Lemma 16.2. Assume r 2. Let X be an r-cube which can be written as an (r 2)-cube of -cartesian squares. Then X is -cartesian. Proof. Use induction on r. If r = 2, then the assertion is trivial. Assume the result holds for some r 1 > 2. Write X as a map Y Z, where Y and Z are (r 1)-cubes, each having the property that it can be written as an (r 3)-cube of -cartesian squares. Then Y and Z are -cartesian. Then X is -cartesian by [G1, prop. 1.6(ii)] (cf. the proof of 4.1).

32 THOMAS G. GOODWILLIE AND JOHN R. KLEIN References [B-K] Bousfield, A. K., Kan, D. M.: Homotopy Limits, Completions and Localizations. (LNM, Vol. 304). Springer 1972 [G1] Goodwillie, T. G.: Calculus II: analytic functors. K-theory 5, 295 332 (1992) [G2] Goodwillie, T. G.: Multiple disjunction for spaces of homotopy equivalences. Preprint, November 29, 1995 [G-K] Goodwillie, T. G., Klein J. R.: Multiple disjunction spaces of smooth embeddings. [Kl1] Klein, J. R.: Poincaré embeddings and fiberwise homotopy theory. Topology 38, 597 620 (1999) [Kl2] Klein, J. R.: Poincaré embeddings and fiberwise homotopy theory. II. Q. J. [Kl3] Math. Oxford 53 319 335 (2002) Klein, J. R.: Poincaré duality spaces. In: Surveys on Surgery Theory, Vol. 1, pp. 135 165 (Ann. of Math. Stud. 145), Princeton Univ. Press 2000 [Q1] Quillen, D.: Homotopical Algebra. (LNM, Vol. 43). Springer 1967 [Q2] Quillen, D.: Higher K-theories. In: Higher algebraic K-theory I, LNM 341, pp. 85 139. Springer Verlag 1973 [W] Waldhausen, F.: Algebraic K-theory of spaces. In: Algebraic and geometric topology, Proceedings Rutgers 1983, LNM 1126, pp. 318 419. Springer 1985 [Wa1] Wall, C. T. C.: Finiteness conditions for CW-complexes. Ann. of Math. 81, 55 69 (1965) [Wa2] Wall, C. T. C.: Poincaré complexes I. Ann. of Math. 86, 213 245 (1967) [Wa3] Wall, C. T. C.: Surgery on Compact Manifolds. Academic Press 1970 Brown University, Providence, RI 02912 E-mail address: tomg@math.brown.edu Wayne State University, Detroit, MI 48202 E-mail address: klein@math.wayne.edu