Configuration spaces on the sphere and higher loop spaces Paolo Salvatore Abstract We show that the homology over a field of the space n nX of free maps f* *rom the n-sphere to the n-fold suspension of X depends only on the cohomology algebra of X and * *compute it explicitly. We compute also the homology of the closely related labelled configuration * *space C(Sn, X) on the n-sphere with labels in X and of its completion, that depends only on t* *he homology of X. In many but not all cases the homology of C(Sn, X) coincides with the homol* *ogy of n nX. In particular we obtain the homology of the unordered configuration spaces on * *a sphere. MSC (2000): 55P48, 55R80, 55S12. Keywords: Configuration space, loop space, homology operation. Introduction In order to compute the homology of the space of based maps n(Y ) = map*(Sn, Y* * ) from the n- sphere to a CW-complex Y , in general one needs to know a great deal of informa* *tion on Y . However, when Y = nX is a n-fold suspension, a classical result by Milgram [9] states t* *hat the homology of n nX depends just on the homology of X. An explicit description is given in [3* *]. This depends upon the existence of a small model for the mapping space. Namely the configuration * *space C(Rn, X) of pairwise distinct points in Rn with labels in X, modulo base point cancellation* *, is homotopy equivalent to n nX when X is connected [6]. In general n nX is the group completion of C* *(Rn, X). Let us turn our attention to the free mapping space nY = map(Sn, Y ). In the* * case n = 1 the homology of X depends only on the homology of X [8]. This depends again on th* *e existence of a small model for the mapping space, the configuration space C(S1, X) of points* * in the circle with labels in X [2], for X connected. This does not extend to all n. For X connecte* *d the configuration space C(Sn, X) is homotopy equivalent to the section space of a bundle over Sn * *with fiber nX [2], obtained from the tangent bundle of the sphere øn by adding a point at inf* *inity to each fiber and smashing it with X. In general this section space is a kind of completion o* *f C(Sn, X) . When n 6= 1, 3, 7, Sn is not parallelizable, so that one does not expect C(Sn, X) to* * be homotopy equivalent to the free mapping space n nX. In this paper we compare the homotopy types of these spaces and we compute th* *eir homology with coefficients in any field. Up to homotopy there are fibrations C(Rn, X) ! * *C(Sn, X) ! nX (for X connected) and n nX ! n nX ! nX, induced by the evaluation at one poi* *nt. Since the basis of both fibrations is a suspension, we can reconstruct the total spaces b* *y means of the clutching functions n-1X x n nX ! n nX. It turns out that the clutching function of th* *e first fibration comes from the action of the little n-discs operad Sn-1 x X x C(Rn, X) ! C(Rn, * *X). This action adds to a `cloud' of labelled points in C(Rn, X) an extra point in the directio* *n parametrized by Sn-1 with label parametrized by X. The same argument works when X is not connected. * *In general we get decompositions both for C(Sn, X) (Theorem 1) and its completion (Theorem 6)* *, that coincide for X connected. The decomposition is compatible with the Snaith splitting (Pro* *position 5). The clutching function induces in homology the so called Browder operation, that ha* *s been computed explicitly for C(Rn, X) in [3]. For example in characteristic 0 and for n = 2 w* *e get the adjoint action of the homology of X on the free Gerstenhaber algebra that it generates. This a* *llows to determine the homology of C(Sn, X), that depends only on the homology of X (Corollary 3).* * In particular for X = S0 we obtain the homology of the unordered configuration spaces on Sn. We w* *rite down an 1 explicit basis of their homology in Theorem 18. The case when n is even and the* * characteristic is odd is not covered by the methods in [1]. Let us consider now the evaluation fibration of the mapping space n nX. In * *this case the clutching function is obtained by twisting the first argument Sn-1xX ! n nX an* *d then applying as before the little n-discs action Sn-1x( n nX)2 ! n nX (Theorem 11) . The tw* *ist is the com- position operation induced by the element c 2 ßn-1( n1Sn) adjoint to the Whiteh* *ead product of the generator ßn(Sn) with itself. This happens exactly because c classifies the fib* *erwise compactification of the tangent bundle of the n-sphere (Lemma 10). Now c goes in homology (up to shift of component) to the Browder operation of* * the identity ' 2 H0( nSn) with itself. This implies that the clutching function of the mappi* *ng space in homology depends only on the cohomology algebra of X (for finite type), and so does the * *homology of n nX (Corollary 12). We show in Example 16 that the homology of n nX does not depen* *d just on the additive homology of X. As an application we write down an explicit basis of th* *e homology of the mapping space nSm , for n m, in Theorems 18 and 19. Let us compare the configuration space (or its completion for X not connected* *) and the mapping space: when c vanishes they are homotopy equivalent. This happens when either X* * is a suspension, or n = 1, 3, 7, or for n odd after inverting the prime 2 (Proposition 9). Moreo* *ver the two spaces have the same homology either mod 2 (Corollary 13), or for X connected and rationall* *y (Corollary 14). In general they do not have the same homology for n even at odd primes (Example* * 16), or rationally when X is not connected (Example 15). We present also a homotopy pullback decomposition of our spaces, induced by d* *ecomposing the n-sphere of the domain as union of two discs (Propositions 20 and 21). In this * *case the twist occurs in the diagram describing the configuration space. We apply the induced sequen* *ce of homotopy groups to show in Example 22 that the configuration space and the mapping space* * are not rationally homotopic for n = 2 and X = CP2, but they have the same homology over all field* *s. It would be interesting to phrase our computations in terms of the higher Hoc* *hschild homology spectral sequence described in 5.9 of [10]. In sections 1,2 and 3 we consider respectively labelled configuration spaces * *on the sphere, the associated spaces of sections, and mapping spaces. In section 4 we compute som* *e examples and we compare the homology of the spaces above. In section 5 we present the homot* *opy pullback decomposition and an application. I am grateful to Sadok Kallel for many helpful conversations. 1 Configuration spaces Let X be a based CW-complex, not necessarily connected. Let n be a positive int* *eger. We recall some definitions from [2]. Let Fk(M) be the space of pairwise disti* *nct k-tuples in a manifold M. We consider the elements of the space Fk(M)x kXk as finite sets of * *points`in M with a label in X. The configuration space C(M, X) is the quotient of the disjoint uni* *on k Fk(M) x kXk under the equivalence relation induced by cancelling points labelled by the bas* *e point of X. Given a closed subset A M, the relative configuration space C(M, A; X) is the quoti* *ent induced by cancelling all points located in A and those labelled by the base point. We identify Sn = Rn [ 1, based at infinity. Let Dn Rn be the unit disc. The* * sequence Dn ! Sn ! (Sn, Dn), for X connected, induces a quasifibration C(Dn, X) ! C(Sn, X) ! * *C(Sn, Dn; X) (Lemma p.178 in [2]). Now nX C(Sn, Dn; X) is a strong deformation retract, * *seen as the subspace of configurations with a single labelled point [7]. We will construct * *a homotopy pushout decomposition of C(Sn, X), by pulling back to the total space_the_decomposition* *_of nX = Sn ^ X induced by splitting Sn as union of the n-discs Dn and En = Sn - Dn. The quasi* *fibration over Dn+ ^ X and En ^ X is homotopically trivial. It remains to glue the two pieces * *together. This idea works even when X is not connected. 2 We recall that C(Dn, X) ' C(Rn, X) is homotopy equivalent to the free algebra* * on the operad of little n-discs generated by X. In particular there is an action of the space* * of two little n-discs Dn(2) by a map æ2 : Dn(2) x C(Rn, X) x C(Rn, X) ! C(Rn, X). Let us restrict th* *is map to the deformation retract Sn-1 Dn(2) in the first factor and to X C(Rn, X), t* *he subspace of configurations consisting of a single labelled point located at the origin, in * *the second factor. We get a map ~ : Sn-1+^X xC(Rn, X) ! C(Rn, X), that we shall call the Browder map. Up * *to deformation ~(z, x, c) is the configuration obtained from the configuration c by adding a p* *oint outside c in direction z with label x. Theorem 1. There is a homotopy pushout ß n Sn-1+^ X x C(Rn, X)_-X x C(R , X) | | | | ~ || || | | ?| ?| C(Rn, X)_________-C(Sn, X), where ~ is the Browder map and ß the projection. Proof.Let us consider the closed deformation retract K of C(Sn, X) containing t* *he configurations with at most one labelled point located outside Dn. The deformation is obtained* * by pushing the particles away from the north pole and deforming slightly the labels towards th* *e base point. Let A, B K be the closed subsets such that the point located outside Dn, if it ex* *ists, is respectively in 2Dn and 2En. Clearly A[B = K. Moreover the inclusion C(Dn, X) A is a homotopy* * equivalence. On the other hand A \ B ~=Sn-1+^ X x C(Dn, X), and B ~=Dn+ ^ X x C(Dn, X). With* * these identifications the map A \ B ! B is induced by the inclusion Sn-1 Dn. Moreov* *er the inclusion A \ B ! A is homotopic to the Browder map. __ |__| From now on we consider homology with coefficients in a field K. We recall th* *e definition of the Browder operation: let Y be a space acted on by the little n-discs operad. Let * *en-12 Hn-1(Dn(2)) be the class represented by the deformation retract Sn-1 Dn(2). Then the Brow* *der operation, for x 2 Hp(Y ) and y 2 Hq(Y ), is defined by [x, y] := æ2*(en-1 x y) 2 Hp+q+n-1(* *Y ). This differs from (5.7) of [3] by the sign (-1)(n-1)p+1. For n = 1 the fundamental class of S0 is {1} - {-1}, so that the Browder oper* *ation is the commutator of the Pontrjagin product. Definition 2. The Browder action is the homomorphism [ , ] = ~* : n-1~H(X) H(C(Rn, X)) ! H(C(Rn, X)). Recall from Theorem 3.1 in [3] that the homology H~(C(Rn, X)) depends functor* *ially on the homology ~H(X). For example in characteristic 0 and for n > 1 the former is the* * free n-algebra on the latter. A n-algebra is a graded variation of a Poisson algebra [11]. In our* * case the Pontrjagin product is the commutative product and the Browder operation is the bracket (up* * to sign). The Browder action for char(K) = 0 is exactly the adjoint action of the space of ge* *nerators ~H(X) on H(C(Rn, X)). In positive characteristic there are additional homology operations generatin* *g H(C(Rn, X)) over H(X), the unstable Dyer-Lashof operations. See (1.1),(1.3) of [3] and Propositi* *on 17. In general the Browder action can be computed explicitly and depends just on * *the homology of X by Thm. 1.2 (5,6,8) and Thm. 1.3 (4) of [3]. For example mod 2 the action * *is trivial on all Dyer-Lashof operations except the top one. Recall that H(C(R, X)) is the free a* *ssociative algebra on ~H(X). The Mayer-Vietoris sequence implies the following corollary. 3 Corollary 3. The homology of C(Sn, X) depends only on the homology of X and is * *the direct sum of the suspended kernel and the cokernel of the Browder action on the homology * *of C(Rn, X). The configuration spaces C(M, X), for a manifold M, are naturally filtered by* * subspaces Ck(M, X), having at most k particles. We denote the subquotients by Dk(M, X) = Ck(M, X)=C* *k-1(M, X) and Dnk(X) = Dk(Rn, X). For example the space Dk(Sn, S0) is the unordered configura* *tion space on the sphere Ck(Sn) = Fk(Sn)= k. The homology H(C(M, X)) splits naturally as sum of H(Dk(M, X)), as k varies. * *Compare Lemma 4.2 p.238 in [3].LThis explains the first assertion of Corollary 3, as H(C(Sn, X)) ~= kH(Fk(Sn); ~H(X) k). The Browder map is compatible with the filtration by number of particles, so * *that we have the following refinement: Corollary 4. The reduced homology of Dk(Sn, X), for k > 1, is the direct sum of* * the suspended kernel and the cokernel of the Browder action n-1~H(X) ~H(Dnk-1(X)) ! ~H(Dnk* *(X)). There is a geometric version of this corollary. Recall that C(M, X) is stably* * homotopy equivalent to the wedge sum of Dk(M, X), indexed over k [2]. The Browder map defines a st* *able map bk : n-1X ^ Dk-1(X) ! Dk(X), by restriction to the relevant wedge summands. For exa* *mple the map b2 is the restriction to the stable summand Sn-1 ^ X ^ X of the quotient map Sn* *-1+^ X ^ X ! Sn-1+^Z2(X ^ X). Proposition 5. The k-th summand of the stable splitting Dk(Sn, X) is stably hom* *otopy equivalent to the cofiber of bk : n-1X ^ Dnk-1(X) ! Dnk(X) for k > 1 and to nX _ X for k* * = 1. Proof.We obtain a homotopy pushout (k) expressing the homotopy type of Ck(Sn, X* *), if we replace, in the diagram of Theorem 1, C by Ck-1 in the top row and by Ck in the bottom r* *ow. For k = 1 this proves the proposition. For k > 1 the cofiber diagram of (k - 1) and (k) g* *ives Dk(Sn, X) as homotopy pushout. The top map ßk of such diagram is a stable retraction, and bk* * is exactly the__ restriction of the left hand side map ~k to the stable kernel of ßk. * * |__| 2 Section spaces We recall that there is a `scanning' map C(Dn, X) ! n nX ~=map*(Dn=Sn-1, Dn=Sn* *-1 ^ X). Let i : X ! n nX be the adjoint of the identity. Roughly speaking, if we scan * *a configuration, we obtain a n-fold loop, sending small discs centered at the configuration points * *to nX ~=Dn=Sn-1^X, by identifying them to the unit disc and applying i to the label of the center.* * The n-fold loop sends everything else to the base point. The scanning map can be extended to C(Sn, X) by using the exponential map of * *the sphere. Let nX ! ø+n(X) ! Sn be the bundle obtained by smashing X fiberwise with the fiber* *wise one-point compactification ø+nof the tangent bundle øn of the n-sphere. Then the scanning* * map goes from C(Sn, X) to the space (ø+n(X)) of sections of the bundle. Up to homotopy the s* *canning maps fit into a diagram from the sequence C(Dn, X) ! C(Sn, X) ! C(Sn, Dn; X) to the evaluation fibration ev n nX ! (ø+n(X)) ___- nX. The scanning maps C(Dn, X) ! n nX and C(Sn, X) ! (ø+n(X)) are homotopy equi* *valences if and only if X is connected. We can define the Browder map for the n-fold loop space n nX, similarly as f* *or the configuration space, by using the action Dn(2)x n nX x n nX ! n nX of the n-discs operad, an* *d restricting the action in the second factor to the subspace X ___i- n nX. 4 If X is connected then the next theorem reduces to Theorem 1. Also the proof * *is an adaptation of the proof of Theorem 1 to the general case. Theorem 6. There is a homotopy pushout ß n n Sn-1+^ X x n nX __-X x X | | | | ~|| || | | ?| ?| n nX ________- (ø+n(X)), where ~ is the Browder map and ß the projection. Proof.Let ev : (ø+n(X)) ! øn(N)+ ^ X ~= nX be the fibration evaluating at the * *north pole N. We construct a map (ff, fi, fl) from the diagram Dn+ ^ X x n nX oe__ Sn-1+^ X x n nX ___- M(~), where M(~) is the mapping cylinder of the Browder map, to the diagram ev-1(Dn+ ^ X) oe__ ev-1(Sn-1+^ X) ___- ev-1(En ^ X). We construct ff so that it covers an automorphism of Dn+ ^ X homotopic to -1 * *^ X. We identify both the southern and the northern hemisphere to the unit disc, by projection f* *rom the opposite pole. This gives a trivialization of the tangent bundle on each hemisphere. The map ff(z ^ x, y) 2 (ø+n(X)) is defined by y 2 Map(Dn, @Dn; nX) on the s* *outhern hemi- sphere. Let OE : (Dn, Sn-1) ! (Sn, 1) be the relative homeomorphism OE(x) = x=* *(1 - |x|). In the northern hemisphere we have the adjoint n-fold loop ff(z ^ x, y)(w) = OE(2(* *w - z=4)) ^ x if |w - z=4| 1=2, and otherwise w goes to the point at infinity. Similarly, on the cylinder I x Sn-1+^ X x n nX M(~), fl(t, z ^ x, y) is de* *fined in the southern hemisphere by y. On the northern hemisphere fl(t, z ^ x, y)(w) = OE(2(w - z(t * *+ 1)=4)) ^ x for |w - z(t + 1)=4| 1=2 and otherwise w goes to the point at infinity. The map f* *l sends the end of the mapping cylinder n nX to ev-1(*) by projection from the north pole. Both ff and fi are homotopy equivalences, because they cover an automorphism * *of the basis and induce homotopy equivalences of the fibers over points of all components. Final* *ly fl is a homotopy equivalence, since the domain has n nX as deformation retract, identified by f* *l to ev-1(*) ' ev-1(En ^ X). By the homotopy invariance of homotopy pushouts the theorem is_pr* *oved._ |__| Also in this case we call Browder action the homomorphism ~* : n-1~H(X) H(* * n nX) ! H( n nX) induced by the Browder map. Corollary 7. The homology of (ø+n(X)) depends only on the homology of X, and i* *s the direct sum of the suspended kernel and the cokernel of the Browder action on the homology * *of n nX. Proof.Recall that H( n nX) is obtained from H(C(Rn, X)), according to the group* * completion theorem, by adding the inverses of the components. In the case n > 1 ß0(C(Rn, X)) = N[ß0(X)], and ß0( n nX) = Z[ß0(X)]. We must * *only compute the Browder action on the inverse of a component a 2 ß0(X) H( n nX). But by t* *he Poisson relation and since the Browder action on the unit is trivial, we obtain [x, a-1* *] = -[x, a]a-2. In the case n = 1 ß0(C(R, X)) and ß0( X) are respectively the free monoid an* *d the free group on the based set ß0(X), and the Pontrjagin product is clear. __ |__| Example 8. If X = S0_S0, then (ø+n(X)) is homotopy equivalent to the section s* *pace of the bundle F3(Rn+1) ! F2(Rn+1) defined by forgetting the third point of a configuration [5* *]. 5 3 Higher loop spaces In some special cases the section space and the mapping space coincide. Proposition 9. The fibration nX ! ø+n(X) ! Sn is trivial, so that the section * *space is homotopic to the ordinary mapping space n nX, in the following cases: 1.When X is a suspension X = Y ; 2.For n = 1, 3, 7; 3.For n odd and away from the prime 2. Proof.1) The tangent bundle of the sphere is trivialized by adding a trivial li* *ne bundle. Thus ø+n( Y ) ~=(øn R)+(Y ) ~= n+1Y x Sn. 2) The spheres S1, S3, S7 are parallelizable. 3) Follows from the following lemma, since 2['n, 'n] = 0 2 ß2n-1(Sn) for n od* *d, where 'n_2_ßn(Sn) is the generator and the brackets denote the Whitehead product. * * |__| Lemma 10. The fibration Sn ! ø+n! Sn with section Sn ! ø+nat infinity is classi* *fied by the adjoint of the Whitehead product ['n, 'n] in ßn-1( n1Sn). Proof.The bundle ø+nis trivial if we forget the section, because it is the sphe* *re bundle of øn R, and its section can be identified to the diagonal : Sn ! Sn x Sn. We will show that the homotopy class of the clutching function c : Sn-1 ! n1* *Sn of the fibration ø+nis exactly the boundary of 'n 2 ßn(Sn) in the long exact sequence associated* * to the evaluation fibration n1Sn ! n1Sn ! Sn. The index 1 denotes the component of degree 1 map* *s. We identify Sn-1 to the equator of Sn. We take the north pole N and the south pole S respec* *tively as base points in the domain and the range. Let H : I x Sn-1 ! SO(n + 1) ! n1Sn be the transformation such that H(t, a) * *rotates by tß the plane generated by a and N, and fixes the orthogonal complement. The clu* *tching function of ø+nis c = H(1, _) : Sn-1 ! n1Sn. Let us identify the unreduced suspension * *of Sn-1 to Sn via [t, a] 7! H(t, a)(N). Then H defines an element in ßn( n1Sn, n1Sn) project* *ing to the generator 'n 2 ßn(Sn), and c = @'n. But by a theorem of Whitehead [12] @('n) is adjoint t* *o_['n, 'n], up to sign convention. |* *__| Let w : Sn-1+^ X ! n nX be the composite of c+ ^ i : Sn-1+^ X ! n1Sn+^ n n* *X and of the composition operation o : n1Sn+^ n nX ! n nX given by o(_, f) = f_. We call the map k(s, x, y) = ~(s, w(s, x), y) the twisted Browder map. Theorem 11. There is a homotopy pushout ß n n Sn-1+^ X x n nX __-X x X | | | | k|| || | | ?| ?| n nX _________- n nX, where k is the twisted Browder map and ß is the projection. Proof.We identify the two hemispheres to the unit disc by orientation preservin* *g diffeomorphisms and we proceed as in Theorem 6. The main difference is that the trivializations* * regard the domain but not the range of the mapping space. Thus the twisted Browder map k replaces ~ in the proof, by comparing the triv* *ializations of the one-point compactified tangent bundle on the two hemispheres with Lemma 10. __ |__| 6 In the next corollary X is supposed to have finite type. Otherwise we must re* *place the expression `cohomology algebra' by `homology coalgebra'. Corollary 12. The homology of n nX with coefficients in a field depends only o* *n the cohomology algebra of X and is the direct sum of the suspended kernel and the cokernel of * *the twisted Browder action . Proof.Let ' 2 H0( n1Sn) be the class of the identity map. By the diagram at p.* * 215 in [3] the Hurewicz homomorphism sends the adjoint Whitehead product in ßn-1( n2Sn) to the* * Browder op- eration [', '] 2 Hn-1( n2Sn), with our sign convention. Therefore, for x 2 H(X)* *, w*(en-1 x) = o*([', ']'-1 x) is the composition product in homology, wherePwe identify x t* *o i*(x) 2 H( n nX). Suppose that the iterated coproduct : H(X) ! H(X3) is (x) = ix0i x00i x0* *00i. Let Ø be the conjugation of the Hopf algebra H( n nX). It depends only on the* * cohomology algebra of X, by induction and by Proposition 1.5 of [3]. P Then the twisted Browder action is k*(x y) = [x, y] + i[x0i, x00i]Ø(x000i)* *y. Compare also Theorem 3.2 (iv) of [4]. __ |__| 4 Examples and applications Corollary 13. At the prime 2 the homology of n nX depends just on the homology* * of X, and is isomorphic to the homology of the section space (ø+n(X)). Proof.The Browder operation [', '] is trivial mod 2. * * |___| Corollary 14. If X is connected then n nX and (ø+n(X)) have the same rational* * homology. Proof.Rationally any suspension is a wedge of spheres, so that nX is rationall* *y a (n + 1)-fold suspension n+1Y , but n n+1Y ' (ø+n( Y )) has the same homology as (ø+n(X))* *_by Corollary 7. |__| The corollary is not true in general, as the following example shows. Example 15. The spaces 2(S2 _ S2) and (ø+2(S0 _ S0)) have not the same ration* *al homology componentwise. Proof.Both spaces have components indexed by Z x Z. The bifiltration of the sec* *tion space = (ø+2(S0 _ S0)) by fiberwise bidegree is compatible with the bifiltration, by n* *umber of particles, of the configuration space C(S2, S0_ S0) of bicoloured particles on the sphere. Co* *mpare also Example 8. The bifiltration of the mapping space M = 2(S2 _ S2) is given by ordinary b* *idegree. Let us denote the generators of ~H0(S0 _ S0) ~H0( 2(S2 _ S2)) by x and y. I* *t turns out from Corollary 7 that the homology group H1( m,n) is the quotient of H1( 2m,n(S2 _ S2)) = Q{[x, x]xm-2yn, [x, y]xm-1yn-1, [y, y]xm yn-2} by the subspace generated by [x, xm-1yn] and [y, xm yn-1]. But these elements * *have respectively coordinates (m-1, n, 0) and (0, m, n-1) with respect to the basis above. It fol* *lows that H1( i,j) ~=Q2 for (i, j) = (1, 1) or (i, j) = (0, 1) or (i, j) = (1, 0) and H1( i,j) ~=Q in a* *ll other cases. On the other hand by Corollary 12 H1(Mm,n) is the quotient of H1( 2m,n(Sn _ S* *n)) by the sub- space generated by [x, xm-1yn] + [x, x]xm-2yn and [y, xm yn-1] + [y, y]xm yn-2,* * that have coordinates (m, n, 0) and (0, m, n). Thus H1(M0,0) ~=Q3 and H1(Mi,j) ~=Q for any (i, j) 6= * *(0, 0). In particular M0,0differs in homology by all components of . __ |__| The next example shows that for n even and at odd primes the homology of the * *mapping space n nX does not depend just on the additive homology of X, so that in particular* * the section space and the mapping space have not the same homology. 7 Example 16. The spaces X = S2 _ S4 _ S6 and Y = CP3 have the same additive homo* *logy , but H9( 2 2X; Z3) ~=Z3 Z3 and H9( 2 2Y ; Z3) ~=Z3. Proof.Let use denote the homology generators of the homology of X = S2 _ S4 _ S* *6 by a1, a2, a3. We must compute the Browder action mod 3 M ~i: H~j(X) Hi-1-j( 2 2X) ! Hi( 2 2X) j for i = 8, 9. For i = 8 the left hand side is generated by a1 [a1, a1] so th* *at ~8 = 0 by The- orem 1.2 (6) in [3]. For i = 9 the left hand side has dimension 6, with genera* *tors a1 a3, a1 a31, a1 a1a2, a2 a2, a2 a21, a3 a1, and the right hand side has dimensi* *on 5, with generators [a1, a3], [a1, a1]a2, [a1, a2]a1, [a2, a2], [a1, a1]a21. By the Poisson relation (Theorem 1.2 (5) in [3]) [a1, a31] = 3[a1, a1]a21= 0 * *mod 3. Thus the image of ~9 has dimension 4, with generators [a1, a3], [a1, a1]a2+ [a1, a2]a1, * *[a2, a2], 2[a2, a1]a1, and H9( 2 2(S2 _ S4 _ S6); Z3) ~=ker(~8) coker(~9) ~=Z3 Z3. Let us consider now the case of the projective space CP3, with homology gener* *ators e1, e2, e3. We must check the twisted Browder action M ki: H~j(CP 3) Hi-1-j( 2 2CP3) ! Hi( 2 2CP3) j for i = 8, 9. By the proof of Corollary 12 k*( e1 y) = [e1, y], k*( e2 y) = [* *e2, y] + [e1, e1]y and k*( e3 y) = [e3, y] + 2[e1, e2]y - [e1, e1]e1y. For the same reason as above k8 = 0. Moreover the image of k9 is generated by t* *he five elements [e1, e3], [e1, e1]e2+ [e1, e2]e1, [e1, e1]e2+ [e2, e2], 2[e1, e2]e1+ [e1, e1]e2* *1, [e1, e3] + 2[e1, e2]e1- [e1, e1]e21. Elementary linear algebra shows that they are linearly independent. Thus k9 i* *s surjective, and H9( 2 2CP3; Z3) ~=ker(k8) coker(k9) ~=Z3. __ |__| We compute next the homology of nSn+k for k 0. We recall the computation o* *f the homology of nSn+k for n > 1. Let X be a space acted on by the operad of little n-discs. For example X = n* *Y or X = C(Rn, Y ). We use the lower index notation for Dyer-Lashof operations and do not consider * *Q0 here. a) Case p = 2: There are operations Qi: Hq(X; Z2) ! H2q+i(X; Z2) for 0 < i < n. (These opera* *tion are also named ,iin [3]). A string QI of symbols Qirepresents the composite operation. I* *t is admissible if the sequence of indices is weakly monotone. The empty string is admissible and repr* *esents the identity. b) Case p odd: There are operations Qi(x) : Hq(X; Zp) ! Hpq+i(p-1)(X; Zp), defined for 0 < i* * < n and when i and q have the same parity. Note that the operations Qikeep the parity and the * *Bockstein operator fi switches parity. An ordered string QI of symbols Qi and fi represents the co* *mposite operation. We say that the operation QI(x) is admissible if 1) the last symbol of the non-empty string QI is Qi, and i has the same parit* *y as the degree of x; 2) the sequence of indices is weakly monotone; 3) the indices of two adjacent Q0s have the same parity; 4) the indices of two Q0s separated by a fi have opposite parity; 5) two adjacent fi do not appear. In particular the empty string is admissible and represents the identity. c) In characteristic 0 only the identity is admissible. 8 Recall that C(Rn, Sk) ' nSn+k for k > 0 [2]. Let ' 2 Hk(C(Rn, Sk)) be the fu* *ndamental class of Sk ' C1(Rn, Sk). Proposition 17. [3] 1) The homology of C(Rn, Sk) is the free commutative graded algebra on the ad* *missible operations QI('), and in addition on the admissible operations QJ([', ']) if n and k have * *the same parity and char(K) 6= 2. 2) The homology of nSn is obtained from C(Rn) = C(Rn, S0) by inverting forma* *lly '. 3) The basis of H(C(Rn)) consisting of products of admissible operations has * *an integral degree that keeps track of the component, corresponding to the number of particles. Th* *is degree is generated by the rules (1) = 0, (') = 1, ([', ']) = 2, (xy) = (x) + (y) and (Qi(x)* *) = (fiQi(x)) = p (x). Now we are ready to state the computation. The next theorem for k > 0 gives a* *lso the homology of nSn+k ' C(Sn, Sk) by Proposition 9(1). Let F denote the graded commutative * *algebra generated by a set of vectors, and let Q be the set of all admissible operations on ' and* * [', '] except the identity. Theorem 18. 1) For n + k odd or for char(K) = 2 H(C(Sn, Sk)) ~=H(C(Rn, Sk)) n+kH(C(Rn, Sk)). 2) For n odd, k odd and char(K) 6= 2 H( nSn+k) ~=H( nSn+k)=([', ']) n+kH( nSn+k)=('). 3) For n even , k even and p = char(K) odd H(C(Sn, Sk)) ~=F(', Q) 'p-1[', ']F('p, Q) n+k[', ']F(', Q) n+kF* *('p, Q). 4) For n even, k even and char(K) = 0 H(C(Sn, Sk)) ~=F(') n+k[', ']F(') n+k{1}. 5) In the case k = 0 the component degree of C(Sn) is obtained from the degre* *e of C(Rn) by the additional rule ( n+kx) = (x) + 1. Proof.We apply Corollary 7. The Browder action is given by [', _]. In case 1) t* *he bracket [', '] is trivial. In all cases the bracket [', x] is trivial if x is any generator of th* *e algebra H( nSn+k) other than '. Namely [', [', ']] = 0 by 1.2(6) of [3], and consequently [', QI(')] = * *0 and [', QJ([', '])] = 0 by 1.2(8) and 1.3(9) of [3]. The action is a derivation with respect to the produc* *t by 1.2(5) of [3]. This completes the computation. __ |__| The computation of the homology of configuration spaces of even spheres at od* *d primes is not deducible from the methods of [1]. It remains to compute the homology of nSn. Theorem 19. 1) For n odd or char(K) = 2 H( nSn) = H( nSn) H(Sn). 2) For n even and char(K) = 0 H( n0Sn) = H(Sn-1) H(Sn) and H( nkSn) ~=H(S2n-* *1) for k 6= 0. 3) For n even and char(K) = p odd H( nkSn) ~=H( n0Sn) H(Sn) if p divides k and H( nkSn) ~=F(Q) n[', ']F(Q) if (p, k) = 1. 9 Proof.Since the bundle ø+nis trivial, there is a homeomorphism (ø+n) ~= nSn, a* *nd we can use either Corollary 7 or Corollary 12. For n odd both the untwisted and the twiste* *d Browder actions are trivial. For n even note that the base point of (ø+n), the section at infi* *nity, is the antipodal map of Sn, that has degree -1. If k is the component of (ø+n) of fiberwise de* *gree k, then k+1~= nkSn. This is compatible with corollaries 7 and 12: the Browder action x 7! ['* *, x] from nkSn to nk+1Sn computes the homology of k+1, but the twisted Browder action from nk-* *1Sn to nkSn ø(x) = [', x] + [', ']'-1x computes the homology of nkSn. They are compatible* * because 'ø(x) = [', x]' + [', ']x = [', 'x]. Cases 1) and 2) are easy. In case 3) the multipl* *ication by 'p induces an isomorphism H( nkSn) ~=H( nk+pSn). If (p, k) = 1 then the composition by a degr* *ee k map induces an isomorphism H( n1Sn) ~=H( nkSn). One concludes the computation by keeping t* *rack of the component degree. __ |__| 5 Homotopy pullbacks Another approach to our spaces is given by homotopy pullbacks, by decomposing t* *he sphere Sn, basis of the evaluation fibration, as union of two n-discs. Proposition 20. The mapping space is the homotopy pullback n nX _____- nX | | | | | c| | | | | ?| c ?| nX ____- n-1 nX, where c is adjoint to the projection Sn-1+^ nX ! nX. Proposition 21. The section space (ø+n(X)) is the homotopy pullback (ø+n(X))____- nX | | | | | c | | | | | ?| v ?| nX ____- n-1 nX, where v : nX ! n-1 nX is adjoint to w : Sn-1+^ X ! n nX. The section and mapping spaces are both total spaces of fibrations with same * *fiber and basis. The mapping space evaluation fibration admits the section given by constant loops. * *Therefore for i > 0 ßi( n nX) = ßi+n( nX) ßi( nX). The section space fibration admits just a stab* *le section, given by the inclusion Sn+^ X = C1(Sn, X) ! (ø+n(X)), since (Sn+^ X) ' ( nX _ X). Example 22. C(S2, CP2) and 2 2CP2 are not rationally homotopy equivalent, but * *they have the same homology over any field. Proof.We show first that the ß05s of the two spaces have different ranks. Ratio* *nally 2CP2 ' S4_S6, so that rk(ß5( 2 2CP2)) = rk(ß7(S4 _ S6)) + rk(ß5(S4 _ S6)) = 1. The fiber square of Proposition 21 induces a long sequence of rational homoto* *py groups . .ß.i(C(S2, CP2)) ___- ßi( 2CP2) ßi( 2CP2) _____c#-v#ßi( 2CP2) ___- ßi-1(C* *(S2, CP2)) . . . Now ß5(S4 _ S6) has rank 0, so that rationally ß5(C(S2, CP2)) is the quotient* * of ß6( 2CP2) = Q{s, u} by the image of c# v#, defined on ß6( 2CP2)2 = Q{s1, s2}. 10 Here s1 and s2 represent the generators coming from the wedge summand S6 ! S4* * _ S6 and s = c#(s1) = c#(s2). Moreover u is the composition S6 ! S4 ! S4 ! (S4 _ S6),* * with the first map adjoint to the Whitehead product ['4, '4] : S7 ! S4. The group [S1+^ CP2, 2 2CP2] splits as [ CP 2, 2 2CP2] [CP 2, 2 2CP2], b* *ecause after two suspensions S1+and S1 _ S0 are equivalent as co-H spaces. Rationally the f* *irst summand ß5( 2 2CP2) = Q{~u} has rank 1 and is detected by the action on ß5 ~=H5. But w : S1+^ CP2 ! 2 2CP2 induces the nontrivial action w*([S1] e2) = [e1,* * e1] where ei2 H2i(CP 2) is the generator, by the proof of Corollary 12. Thus w correspond* *s in the decomposition above to ~u i, with i adjoint to the identity. By taking adjoints, v : 2CP2 !* * 2CP2 represents the sum in the group [ 2CP2, 2CP2] of the injection c : 2CP2 ! 2CP2 and th* *e composition 2CP2 ! S6 ___u- 2CP2. It follows that v#(s2) = s + u, and ß5(C(S2, CP2)) has rank 0. The second assertion follows from Corollary 13 and the fact that away from th* *e prime 2 2CP2 ' S4 _ S6. __ |__| This shows that the rational homotopy type of C(S2, X) does not depend just o* *n the rational homology of X, by taking X = CP2and X = S2 _ S4. Dipartimento di Matematica, Universit`a di Roma öT r Vergata", Via della Ricerca Scientifica 1, 00133 Roma, Italy e-mail: salvator@mat.uniroma2.it References [1]C.-F. Boedigheimer, F.R. Cohen, L. Taylor, Homology of configuration spaces,* * Topology 28 (1989), 111-123. [2]C.-F. Boedigheimer, Stable splittings of mapping spaces, LNM 1286, 1987, 174* *-187. [3]F.R. 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