ON THE HOMOTOPY INVARIANCE OF CONFIGURATION SPACES MOKHTAR AOUINA AND JOHN R. KLEIN Abstract. For a closed PL manifold M, we consider the configuration space F (M, k) of ordered k-tuples of distinct points in M. We show that a suitable iterated suspension of F (M, k) is a homotopy invariant of M. The number of suspensions we require depends on three parameters: the number of points k, the dimension of M and the connectivity of M. Our proof uses a mixture of embedding theory and fiberwise algebraic topolog* *y. 1. Introduction For a closed PL manifold M and an integer k 2, we will consider the configuration space F (M, k) := {(x1, ..., xk)| xi2 M and xi6= xj fori 6= j} . A fundamental unsolved problem about these spaces concerns their homotopy invariance: when M and N are homotopy equivalent, is it true that F (M, k) and F (N, k) are homotopy equivalent? Here is some background. It is known that the based loop space F (M, k), is a homotopy invariant (see Levitt [L ]). When M is smooth, the cohomol- ogy of F (M, k) with field coefficients has been intensively studied (see e.g., Bödigheimer-Cohen-Taylor [B-C-T ]). When M is a smooth projective variety over C, Kriz [Kr ] has shown that the rational homotopy type of F (M, k) de- pends only on the rational cohomology ring of M. These results indicate that if homotopy invariance fails, a counterexample will be difficult to come by. When k = 2 we have F (M, 2) = M x M - is the deleted product. Even in this instance, the homotopy invariance question is still not settled (althou* *gh partial results are known; see Levitt [L ]). The purpose of this paper is to show that a suitable iterated suspension of F (M, k) is a homotopy invariant. The bound on the number of suspensions we need to take depends on three parameters: the number of points, the dimension of M and the connectivity of M. ____________ Date: October 30, 2003. The second author is partially supported by NSF Grant DMS-0201695. 2000 MSC. Primary 55R80; Secondary 57Q35, 55R70. 1 2 MOKHTAR AOUINA AND JOHN R. KLEIN For an unbased space Y , we define its j-fold suspension jY := (* x Sj- 1) [ (Y x Dj) , where the union is amalgamated along Y x Sj- 1(up to homotopy, jY is the join of Y and Sj- 1). For integers d, k 3 and r 0, define ff(k, d, r) := (k - 2)d - r + 2 . When k = 2 and r 3, or when d 2, we set ff(k, d, r) := 0. Our main result is Theorem A. Let M and N be homotopy equivalent closed PL manifolds of dimension d. Assume M is r-connected for some r 0. Then there is a homotopy equivalence ff(k,d,r)F (M, k) ' ff(k,d,r)F (N, k) , Remark. (1). Cohen and Taylor (unpublished manuscript) prove by very dif- ferent methods that the configuration spaces of smooth manifolds are stable homotopy invariant. In their work the bound on the number suspensions re- quired to achieve homotopy invariance is significantly weaker. Nevertheless, an advantage of their approach is its applicability to other kinds of configuration spaces. For example, their results apply as well to the unordered configuration spaces of a smooth manifold. We are unable to analyse the latter using our methods. We now single out two corollaries of our main result. Assume in what follows that M is a connected closed PL manifold. Corollary B. The suspension spectrum 1 F (M, k)+ is a homotopy invariant of M. The second corollary extends the work of Levitt [L ], who considered only the case r = 2. Corollary C (k = 2). If M is r-connected for some 0 r 2, then 2-rF (M, 2) is a homotopy invariant of M. Conventions. We work in the category Top of compactly generated spaces. A non-empty space is always (-1)-connected. A non-empty space is 0-connected if it is path connected. It is r-connected for r > 0 if it is path connected and its homotopy groups (with respect to a choice of basepoint) vanish in degrees r. A map A ! B of spaces (with B non-empty) is r-connected if for any choice of basepoint in B, the homotopy fiber with respect to this choice of basepoint is an (r- 1)-connected space. A weak (homotopy) equivalence is an 1-connected map. If two spaces A and B are related by a chain of weak equivalences, we will often indicate it by writing A ' B. ON THE HOMOTOPY INVARIANCE OF CONFIGURATION SPACES 3 2.Fiberwise suspension Let A ! X be a map of spaces. Define TopA!X to be the category of spaces "between A and X." Specifically, an object is a space Y and a choice of factorization A ! Y ! X. A morphism is a map of spaces which is compatible with their given factorizations. Call a morphism a weak equivalence if it is a weak homotopy equivalence of underlying spaces. We use the notation Top=X for Top;!X . If Y 2 Top=X is an object, define its (unreduced) j-fold fiberwise suspension by jXY := (Y x Dj) [ (X x Sj-1) , where the union is amalgamated over Y xSj-1. With respect to the first factor projection map X x Sj- 1! X, we get a functor jX:Top=X ! TopXx Sj-1!X . Lemma 2.1. Let Y and Z be objects of Top=X whose underlying spaces are path connected and have the homotopy type of CW complexes. Assume for some j 0 that jXY and jXZ are weak equivalent objects. Then there is a weak equivalence of spaces jY ' jZ . Proof. The statement is obviously true for j = 0, so we will assume that j > 0. Moreover, we may assume that we are given a weak equivalence jXY !~ jXZ. For any object T 2 Top=X , we have a cofibration sequence of spaces X x Sj-1 ! jXT ! j(T+ ) , where we use the fact that j(T+ ) means T x Dj with T x Sj- 1collapsed to a point. Using this cofiber sequence for both Y and Z, we get a commutative diagram jXY ---! j(Y+ ) ? ? ' ?y ?y jXZ ---! j(Z+ ) which is also homotopy pushout. It is well-known that cobase change pre- serves weak equivalences (see e.g., Hirschhorn [H ]), so it follows that the map j(Y+ ) ! j(Z+ ) is a weak equivalence. Choose basepoints for Y and Z. Since j > 0, we have j(Y+ ) ' ( jY ) _ Sj and similarly j(Z+ ) ' ( jZ) _ Sj. It follows that there is a weak equivalence ( jY ) _ Sj ' ( jZ) _ Sj . 4 MOKHTAR AOUINA AND JOHN R. KLEIN Because Y and Z are connected, we have that jY and jZ are j-connected. Using Lemma 2.2 below, we conclude that the composite project j jY include----!( jY ) _ Sj ' ( jZ) _-Sj---!( Z) is a weak equivalence. Lemma 2.2. Let U and V be j-connected spaces with j 0. Assume U and V are equipped with non-degenerate basepoints. Assume h: U _ Sj ! V _ Sj is a weak equivalence. Then the composition project g :U -include---!U _-Sjh--!V _ Sj ----! V is also a weak equivalence. Proof. Without loss in generality we can assume that U and V are CW com- plexes with no cells in positive dimensions j. By cellular approximation, we may also assume that h is a cellular map. Then h preserves j-skeleta, so there is a commutative diagram Sj - --! U _ Sj ? ? h|Sj?y ?yh , Sj - --! V _ Sj and it is straightforward to check that the left vertical map is a homotopy equivalence. We infer that the map U ! V obtained by taking cofibers hori- zontally is also a weak equivalence. But this map coincides with g. 3. Embeddings up to homotopy Let K be a space. Write dim K k if K is up to homotopy a cell complex of dimension k. We say that K is homotopy finite if it is homotopy equivalent to a finite cell complex. Let M be a PL manifold of dimension d, possibly with boundary. Fix a map f :K ! M, in which K is a homotopy finite space. Definition 3.1. An embedding up to homotopy of f is a pair (N, h) in which o N denotes a compact codimension zero PL submanifold of the interior of M, and o h: K ! N is a homotopy equivalence such that composition K !h N M is homotopic to f. ON THE HOMOTOPY INVARIANCE OF CONFIGURATION SPACES 5 A concordance of embeddings up to homotopy (N0, h0) and (N1, h1) of f consists of o an embedded PL h-cobordism (W, N0, N1) (M x I, M x 0, M x 1) , where @W = N0[ @1W [ N1 and @1W is the internal part of @W , (i.e., W \ M x {i} = Ni for i = 0, 1); o a homotopy equivalence H :(Kx I, Kx 0, Kx 1) ~! (W, N0, N1) which factors the map fx id up to homotopy. We remark that our definition of embedding up to homotopy differs from the one of Stallings and Wall in that we do not work with simple homotopy equiv- alences. Our notion of concordance accounts for this distinction (Stallings and Wall use s-cobordisms in their notion of concordance); our set of concordance classes coincides with theirs when dim K d - 3. Theorem 3.2 (Stallings [St], Wall [Wa1 ]). Assume dim K k d- 3. If f :K ! M is (2k- d+ 1)-connected, then f embeds up to homotopy. Further- more, any two embeddings up to homotopy of f are concordant whenever f is (2k- d+ 2)-connected. 4. Decompression Let (N, h) be an embedding up to homotopy of f :K ! M. If C denotes the closure of the complement of N inside M, then C is an object of Top@M M . Definition 4.1. The object C 2 Top@M M is called the complement of (N, h). By considering the inclusion M x 0 M x Dj, and taking a compact regular neighborhood of N in M x Dj, we have an associated embedding up to homotopy of the composite f j fj: K ! M = M x 0 M x D . Denote this embedding up to homotopy by (Nj, hj), where Nj ~=N x Dj and the homotopy equivalence hj is identified the composite K !h N N x Dj. This new embedding up to homotopy is called the j-fold decompression of (N, h). Its complement has the structure of an object of TopMx Sj-1 Mx Dj. 6 MOKHTAR AOUINA AND JOHN R. KLEIN However, to avoid technical problems, we will henceforth regard the com- plement as a space over M by projecting away from the Dj factor. That is, we will think of the complement as an object of TopMx Sj-1!M . Lemma 4.2 (Compare [Kl2 , x2.3]). Assume that M is closed. Then the com- plement of (Nj, hj) is weak equivalent to the object jMC . Proof. The regular neighborhood Nj can be chosen as N x D1=2 M x Dj, where D1=2 Dj is the disk of radius 1=2. The complement of (Nj, hj) is then (M x Dj) - int(N x D1=2) = C x D1=2 [ M x D[1=2,1], where D[1=2,1]denotes the annulus consisting of points in Dj whose norm varies between 1=2 and 1. The above union is amalgamated over C x @D1=2. The subspace of the complement given by (C x D1=2) [ (M x D1=2) is evidently isomorphic to jMC. The inclusion map of this subspace is, up to isomorphism, a morphism of TopMx Sj-1!M . Furthermore, this inclusion is a weak homotopy equivalence of underlying spaces. 5. The suspended complement Proposition 5.1. Assume f :Kk ! Md is an r-connected map, where M is a closed connected PL manifold of dimension d, and k d- 3. Suppose that f has two embeddings up to homotopy (N, h) and (N0, h0) with respective complements C and C0. Then there is a homotopy equivalence, jC ' jC0, where j = max (2k - d - r + 2, 0). Proof. By the Stallings-Wall theorem, with j = max (2k - d - r + 2, 0), we see that the j-fold decompressions of (N, h) and (N0, h0) are concordant. Using Lemma 4.2 we infer that there is a weak equivalence of objects jMC ' jMC0. By Lemma 2.1, we conclude jC ' jC0. 6. Proof of Theorem A Suppose that M and N are homotopy equivalent r-connected (r 0) closed PL manifolds of dimension d. With appropriate modifications, we will argue along the lines of Levitt's strategy for showing F (M, 2) ' F (N, 2) when M and N are 2-connected (see [L ]). ON THE HOMOTOPY INVARIANCE OF CONFIGURATION SPACES 7 Case 1: d 2. By the classification of low dimensional manifolds, M and N are PL homeomorphic. It follows that F (M, k) and F (N, k) are homeomorphic for all k. Case 2: d > 2. Let fatk(M) Mxk denote the fat diagonal. This subpolyhedron is the space of k-tuples of points of M such that at least two entries in the k-tuple coincide. By choosing a regular neighborhood, we obtain an embedding up to homo- topy of the inclusion fatk(M) Mxk . Its complement C is weak equivalent to F (M, k) when the latter is considered as an object of Top=Mxk . Denote this embedding up to homotopy by (V, h). Note that that have an associated codimension one manifold splitting given by (Mxk ; V, C; @V ) . Repeat this procedure for the fat diagonal of N in Nxk to get an em- bedding up to homotopy of the inclusion fatk(N) Nxk . Call the latter embedding up to homotopy (W, h0). Its complement C0 is identified with F (N, k) 2 Top=Nxk . Thus we have a codimension one splitting (Nxk ; W, D; @W ) . The next step is to choose a homotopy equivalence g :M !~ N. The k-fold product of g with itself gives another homotopy equivalence gk: Mxk !~ Nxk . We can therefore use gk to form another triad (W [ C; W, C; @W ) together with a homotopy equivalence _ :Nxk ! W [ C (note that we are using C instead of D). The latter triad is codimension one Poincar'e dual- ity splitting. According to the Browder-Casson-Sullivan-Wall theorem [Wa2 , Th. 12.1], such splittings can be made into manifold splittings: there exists codimension one manifold splitting (Nxk ; W 0, C0; @W 0) and a homotopy equiv- alence of triads OE: (Nxk ; W 0, C0; @W 0) ~! (W [ C; W, C; @W ) such that OE: Nxk ! W [ C is homotopic to _. These data describe another embedding up to homotopy of the inclusion fatk(N) ! Nxk with the property that its complement is identified with F (M, k) up to homotopy equivalence. Summarizing thus far, we have two embeddings up to homotopy of the inclusion fatk(N) ! Nxk , one whose complement is identified with F (N, k) and the other whose complement is identified with F (M, k). The next step of the argument is to verify the hypotheses of Proposition 5.1. One checks by elementary means that dim fatk(N) (k- 1)d. As d > 2, the hypothesis (k- 1)d kd - 3 is satisfied. Furthermore, the inclusion map 8 MOKHTAR AOUINA AND JOHN R. KLEIN fatk(N) ! Nxk is r-connected (recall that r is the connectivity of N). Hence, applying 5.1, we infer jD ' jC0, where j = max (2(k- 1)d - kd - r + 2, 0). It is then straightforward to check that j = ff(k, d, r). Finally, we recall that D ' F (N, k) and C0 ' F (M, k). With respect to these identifications, we get ff(k,d,r)F (M, k) ' ff(k,d,r)F (N, k) . This concludes the proof of Theorem A. References [B-C-T]Bödigheimer, C.-F., Cohen, F., Taylor, L.: On the homology of configura* *tion spaces. Topology 28, 111-123 (1989) [H] Hirschhorn, P. S.: Model categories and their localizations. (Mathematic* *al Surveys and Monographs, Vol. 99). Amer. Math. Soc. 2003 [Kl2] Klein, J. R.: Poincar'e embeddings and fiberwise homotopy theory. Topolo* *gy 38, 597-620 (1999) [Kr] Kriz, I.: On the rational homotopy type of configuration spaces. Ann. of* * Math. 139, 227-237 (1994) [L] Levitt, N.: Spaces of arcs and configuration spaces of manifolds. Topol* *ogy 34, 217-230 (1995) [St] Stallings, J. R.: Embedding homotopy types into manifolds. 1965 unpubli* *shed paper (see http://math.berkeley.edu/~stall for a TeXed version) [Wa1] Wall, C. T. C.: Classification problems in differential topology_IV. Thi* *ckenings. Topology 5, 73-94 (1966) [Wa2] Wall, C. T. C.: Surgery on Compact Manifolds. (Mathematical Surveys and * *Mono- graphs, Vol. 69). Amer. Math. Soc. 1999 Mokhtar Aouina, Dept. of Mathematics, Wayne State University, Detroit, MI 48* *202 E-mail address: aouina@math.wayne.edu John R. Klein, Dept. of Mathematics, Wayne State University, Detroit, MI 482* *02 E-mail address: klein@math.wayne.edu