A remarkable DG-module model for configuration spaces

A REMARKABLE DG-MODULE MODEL F OR CONFIGURA TION SP A CES P ASCAL LAMBRECHTS AND DON ST ANLEY Abstract. Let M be a simply- connect ed closed manifold and consider the (ordered) configuration space of k points in M , F ( M , k ). In this paper we construct a co mmutativ e differen tial graded algebra which i s a p otent ial can- didate for a mo del of the rational homotopy type of F ( M, k ). W e pr ov e that our mo del it is at least a Σ k -equiv arian t differential graded mo del. W e also study Lefsc hetz duality at the lev el of cochains and describ e equi- v arian t models of the complemen t of a union of polyhedra in a closed manifold. 1. Introduction Let M b e a closed s imply- connected triangulable ma nifold of dimension m . The (ordered) config uration spac e of k p oints in M is the space F ( M , k ) := { ( x 1 , · · · , x k ) ∈ M k : x i 6 = x j for i 6 = j } . An in teresting problem is whether the homotopy t ype of that c o nfiguration space depe nds only on the homo topy type of M . Longoni and Salv atore [16] hav e discov- ered a n example of tw o homoto py equiv alent manifolds who se co nfiguration s paces of tw o p oints ar e not homotop y e q uiv alen t. Their examples a r e no n-simply con- nected. By contrast, a g e neral p ositio n ar gument implies that fo r a 2-co nnected closed manifold the c onfiguration space of tw o po ints depends only on the homo- topy type o f the ma nifo ld. More genera lly we hav e prov ed in [14] that the ra tional homotopy t yp e of F ( M , 2) depe nds only on the rationa l ho motopy type of M , under the 2-connectivity hypothesis, a nd we hav e build an explic it mo del (in the sense of Sulliv an) of that configuratio n space o ut o f a model of M . The goal of the present pap er is to exhibit a promising candidate for the mo del of the ra tional homoto py type o f the F ( M , k ). T o explain this, first recall the Sulliv an functor A P L : T o p → CDGA where CDGA is the categ ory o f commutativ e differen tial graded a lgebras. The main feature of this functor is that the r ational ho motopy type of a simply-co nnected space of finite type, X , is enco ded in a ny CDGA quasi- isomorphic to A P L ( X ). Such a CDGA is called a CDGA-mo del of X . 1991 Mathematics Subje ct Classific ation. 55R80 , 55P62. Key wor ds and phr ases. Poincar ´ e dualit y . Lefsch etz dualit y . Sulliv an mo del. Configuration spaces. The authors gratefully ack nowledge supp ort by the Institute Mittag-Leffler (Djurs holm , Swe- den). P . L. i s Chercheur Qualifi´ e au F.N. R.S.. 1 2 P ASCAL LAMBRECHTS AND DON ST ANLEY In [1 3] we hav e pro ved that an y simply-connected manifold M admits a CDGA mo del, ( A , d ), such that A is a Poincar´ e duality algebr a of dimens io n m = dim M . W e can then define a diagonal class ∆ := X λ ( − 1) deg( a λ ) a λ ⊗ a ∗ λ ∈ A ⊗ A where { a λ } is a bas is of A and { a ∗ λ } is the Poincar´ e dual basis. In the present paper we descr ib e a CDGA (1.1) F ( A, k ) :=  A ⊗ k ⊗ E( g ij : 1 ≤ i < j ≤ k ) (Arnold and symmetry relations) , d ( g ij ) = π ∗ ij (∆)  where E( g ij ) is an exterio r a lg ebra on  k 2  generator s g ij of degree m − 1, π ∗ i ( a ) = 1 ⊗ i − 1 ⊗ a ⊗ 1 ⊗ k − i ∈ A ⊗ k and π ∗ ij ( a ⊗ b ) = π ∗ i ( a ) · π ∗ j ( b ) (see Definition 3.4 for a complete definition.) When k = 2, F ( A, 2) is w eakly equiv alent to the CDGA model of F ( M , 2) built in [14, Theor em 5.6], and when M is a co mplex pro jective v ariety then F ( H ∗ ( M ; Q ); k ) is equiv alen t to the F ulto n- MacPhers o n-Kriz CDGA-model of F ( M , k ) built in [9] and [12]. W e are not a ble to pr ov e in general that for k ≥ 3 , F ( A, k ) is a CDGA-mode l of F ( M , k ) but at least we can pr ov e that it is an equiv ariant DG- mo dule mo del o f it. More prec isely the inclusion F ( M , k ) ֒ → M k and Kunneth quasi-isomo rphism induce an A P L ( M ) ⊗ k -mo dule structure on A P L ( F ( M , k )). Sup- po se given q uasi-isomo r phisms of CDGA, A ≃ ← R ≃ → A P L ( M ). Our main result (Theorem 10 .1) states that A P L ( F ( M , k )) and F ( A, k ) are weakly equiv alent R ⊗ k - DGmodules , even Σ k -equiv arianly where Σ k is the symmetric gr oup on k letters acting by p ermutation of the factor s. Our pr o of go es through a n “ equiv ariant co chain-level Lefschetz dualit y theo- rem for a system o f subp olyhedra in a close d manifold.” In more detail, cla ssical Lefschetz duality determines H ∗ ( W r X ) fro m the ma p H ∗ ( X ) → H ∗ ( W ) when X is a subpo ly hedron of a clo sed oriented manifold W . In [1 5] we hav e studied Lefschetz duality at the level of mo dels instead of homology . In this pap er we generalize this further b y co nsidering X as a union of a finite family of subp oly- hedra { X e ֒ → W } e ∈ E . The idea is tha t Lefsch etz dua lit y gives a weak e quiv a- lence be tw een C ∗ ( W r ∪ e ∈ E X e ) and the mapping cone of the dual of the map C ∗ ( W ) → C ∗ ( ∪ e ∈ E X e ). On the other ha nd a g e ne r alized Ma yer-Vietoris theor e m gives a weak eq uiv alence betw een C ∗ ( ∪ e ∈ E X e ) and a chain complex built out of the chain complexes C ∗ ( ∩ e ∈ γ X e ) for non empty subsets γ ⊂ E . When a discrete gr oup G acts on the manifo ld W pr eserving in a certain sense the sys tem { X e ֒ → W } e ∈ E , all these weak equiv alences can be choosen to be equiv ariant. This generalized Lefsc hetz dualit y c a n be applied to the system of pa r tial diag- onals ∆ ij = { ( x 1 , · · · , x k ) ∈ M k : x i = x j } so that F ( M , k ) = M k r ∪ 1 ≤ i k γ k , ǫ r otherwise. Pr o of. In the first case this map is the dual of a signed p ermutation of A ⊗ r +1 follow ed by a multiplication o f t wo adjacent factor s. An a rgument ana lo gous to that of Lemma 10.2 b y e v aluation on a ba sis o f A ⊗ r +1 implies the formula. In the second case, k γ r e k = k γ k and A ( γ → γ r e ) is the iden tity map.  L emma 10.4 . L et γ ∈ Γ b e a non r e dundant gr aph. Then the differ ential D in the total c ofibr e ( 10.1 ) satisfies D  s − mk y γ ǫ k γ k  = ( − 1) mk X e ∈ γ s − mk ( − 1) p os( e : γ ) y γ r e ∆( e ) ǫ k γ k +1 . 20 P ASCAL LAMBRECHTS AND DON ST ANLEY Pr o of. Use the form ula of D in Definition 7.2 and Lemma 10.3.  The following lemma serves to define sig ns λ ( γ ) = ± 1 that app ears in the defi- nition of Φ in Lemma 10.6. The formula below is exactly the one needed to ma ke Φ commute with the differential (see Lemma 10.1 0.) L emma 10.5 . Ther e exists a map λ : Γ → {− 1 , +1 } such that λ ( ∅ ) = 1 and for e ach non r e dundant gr aph γ ∈ Γ and e ∈ γ λ ( γ ) = − ( − 1 ) m (p os( e : γ )+ k γ k ) ν ( γ , e ) λ ( γ r e ) . Pr o of. Set R ( γ , e ) := − ( − 1) m (p os( e : γ )+ k γ k ) ν ( γ , e ) so that the equation of the state- men t is λ ( γ ) = R ( γ , e ) · λ ( γ r e ). F or a non redundant graph γ we define λ ( γ ) by induction on | γ | using this equation but we need to pro ve that it is indep endent o f the choice of the edge e ∈ γ . F or this it is enough to s how tha t if e 1 and e 2 are tw o distinct edg es in γ then R ( γ , e 1 ) · R ( γ r e 1 , e 2 ) = R ( γ , e 2 ) · R ( γ r e 2 , e 1 ) , which is equiv alent to (10.2) ν ( γ , e 1 ) ν ( γ r e 1 , e 2 ) = ( − 1) m ν ( γ , e 2 ) ν ( γ r e 2 , e 1 ) . Set r = k γ k . Using Lemma 10.3 we compute (# A ( γ r e 1 → γ r { e 1 , e 2 } )) ((# A ( γ → γ r e 1 )) ( ǫ r )) = = ν ( γ , e 1 )∆( e 1 ) ((# A ( γ r e 1 → γ r { e 1 , e 2 } )) ( ǫ r +1 )) = = ν ( γ , e 1 )∆( e 1 ) ν ( γ r e 1 , e 2 )∆( e 2 ) ǫ r +2 . A s imila r computation gives # A ( γ r e 2 → γ r { e 1 , e 2 } ) (# A ( γ → γ r e 2 )( ǫ r )) = ν ( γ , e 2 )∆( e 2 ) ν ( γ r e 2 , e 1 )∆( e 1 ) ǫ r +2 . Since # A is a functor, the last tw o expressio ns a r e equal and this implies Equatio n (10.2) b ecause ∆( e 1 )∆( e 2 ) = ( − 1) m ∆( e 2 )∆( e 1 ).  L emma 10.6 . Ther e exists a un ique A ⊗ k -mo dule map Φ : s − mk T otCof (# A ) → F ( A, k ) such that for γ ∈ Γ Φ  s − mk y γ ǫ k γ k  = λ ( γ ) g γ . Pr o of. The facto r s − mk y γ # A ⊗k γ k is a free A ⊗k γ k -mo dule g enerated by s − mk y γ ǫ k γ k . Its A ⊗ k -mo dule structure is induced by an algebr a map A ⊗ k → A ⊗k γ k obtained as a p ermutation follo wed by iterated m ultiplications. The fact that Φ( s mk y γ ǫ k γ k ) = λ ( γ ) g γ can b e extended to a A ⊗ k -mo dule map is a co nsequence of the symmetry relations π ∗ i ( a ) g ij = π ∗ j ( a ) g ij in F ( A, k ).  Notice that if γ is a redundant graph then Φ  s − mk y γ ǫ k γ k  = 0. The three next le mmas esta blish the equiv ar iance of Φ. L emma 10.7 . L et γ ∈ Γ and σ ∈ Σ k . We have the fol lowing e quation in the total c ofibr e ( 10.1 ) σ ·  s − mk y γ · ǫ k γ k  = s gn( σ : γ ) (sg n( σ )sgn ( σ : π 0 ( γ ))) m s − mk y σ · γ · ǫ k σ · γ k MODELS F OR CONFIGURA TION S P ACES 21 Pr o of. The facto r s gn( σ : γ ) is the sign coming from the actio n on y γ in the cubical diagr am a s in Eq uation (8.1), sgn( σ ) m is the or ientation-t wisting, and sgn( σ : π 0 ( γ )) m is the Ko szul sign o f the p er m utation A ⊗k γ k ∼ = → A ⊗k σ · γ k on an element of top degre e .  F or 1 ≤ p ≤ k − 1 a nd for an edge e ∈ E or a graph γ ∈ Γ w e se t η p e := ( ( − 1) m if e = ( p, p + 1), +1 otherwise. η p γ := ( ( − 1) m if ( p, p + 1) ∈ γ , +1 otherwise. L emma 10.8 . L et 1 ≤ p ≤ k − 1 , c onsider the tr ansp osition τ = ( p, p + 1) ∈ Σ k , let γ ∈ Γ b e a non re dundant gr aph and let e ∈ γ . Then ν ( γ , e ) ν ( τ · γ , τ · e ) = η p e (sgn( τ : π 0 ( γ )) sgn( τ : π 0 ( γ r e ))) m ; (10.3) λ ( γ ) λ ( τ · γ ) = η p γ ( − sgn( τ : γ ) s gn( τ : π 0 ( γ ))) m ; (10.4) τ · g γ = η p γ sgn( τ : γ ) m − 1 g τ · γ . (10.5) Pr o of. (10.3) Since the different ial D on s − mk T otCof (# A ) is equiv ar iant we hav e τ .D ( s − mk y γ · ǫ k γ k ) = D ( τ .s − mk y γ · ǫ k γ k ) . Develop both sides of this equation using Lemma 10.4 and Lemma 8 .2. The sig n η p e comes fr om the fact that τ · ∆(( p, p + 1)) = ( − 1) m ∆(( p, p + 1)). (10.4) By induction on the n umber of edg es | γ | using Lemmas 10.5 and 8.2 and the previo us formula. (Hin t: in the induction cho o se the edge e ∈ γ to b e ( p, p + 1) when it belongs to γ .) (10.5) The sign η p γ comes from g p +1 ,p = ( − 1) m g p,p +1 and the other sign is the Koszul sig n of the rea rrang ment of the g e which are of degree m − 1 .  L emma 10.9 . Φ is Σ k -e quivariant. Pr o of. It is enough to chec k the equiv a riance for transp ositions τ = ( p, p + 1) o f adjacent vertices applied to the g enerator s s − mk y γ · ǫ k γ k . If γ is non re dunda nt it is a c omputation using Lemmas 10.7 a nd 10.8. If γ is redunda nt then the same is true for τ · γ and the images by Φ of the corresp onding genera tors ar e 0.  L emma 10.10 . Φ c ommutes with the differ entials. Pr o of. Since Φ is an A ⊗ k -mo dule map b etw e e n A ⊗ k -DGmo dules, it is enough to chec k this o n the gener a tors s − mk y γ · ǫ k γ k . F o r a non redundant graph it is a co mpu- tation using Lemmas 10.4 a nd 10.5. T o finis h the proo f we establish the following: Claim: if γ is a redunda nt gra ph then Φ( D ( s − mk y γ ǫ k γ k )) = 0 . F or the sake o f the pro o f we define an l -cycle in a graph γ as a subset o f edges { ( i 1 , i 2 ) , ( i 2 , i 3 ) , . . . , ( i l − 1 , i l ) , ( i 1 , i l ) } . A gra ph γ is redundan t if and only if it co n- tains some l -cycle, with l ≥ 3, and then g γ = 0 . Notice that when γ contains more than one cycle, in other words when γ r e is still r e dunda nt for any e dg e e ∈ γ , then the claim is o bvious. So from now on w e supp os e that γ contains exa ctly one cycle. The claim is easy for the gra ph γ 123 := { (1 , 2) , (1 , 3) , (2 , 3) } using the Arnold re- lation in F ( A, k ) (hint: to c o mpare the different signs λ ( γ 123 r e ), use (1 0.4) in Lemma 10.8.) B y an induction on the num ber o f edges one deduces the claim for any graph containing γ 123 and no other cycle. By the equiv ariance of Φ this implies the r esult for an y graph c ontaining a 3 - cycle. 22 P ASCAL LAMBRECHTS AND DON ST ANLEY Finally o ne prov e s the result for an y g raph containg an l - cycle, for l ≥ 4 b y induc- tion on l . Indeed if γ con tains the l -cycle (1 , 2 ) , · · · , ( l − 1 , l ) , (1 , l ) then consider the graph ˆ γ := γ ∪ { (1 , 3) } . The terms of D ( s − mk y ˆ γ ǫ k ˆ γ k ) c o ntains one term indexed by γ a nd o ther terms indexed by a gra ph containg a cycle of length < l or more than one c ycle. Using that D 2 = 0 and the inductive hypothesis one deduces the claim.  L emma 10.11 . Φ is a quasi-isomorp hism. Pr o of. Let Γ 0 ⊂ Γ b e the subset co nsisting of graphs of the form { ( i 1 , j 1 ) , . . . , ( i l , j l ) } with 1 ≤ i 1 < · · · < i l ≤ k all distinct and i s < j s ≤ k for s = 1 , . . . , l . Co nsider the inclus ion of ch ain complexes ι : s − mk  ⊕ γ ∈ Γ 0 y γ · #( A ⊗k γ k )  ֒ → s − mk  ⊕ γ ∈ Γ y γ · #( A ⊗k γ k )  . An arg ument completely analo gous to that of [8 , Prop osition 2 .4] (passing to the duals) shows that ι is a quasi-iso mo rphism. Since Φ ι is an isomorphism we deduce that Φ is a quasi-isomor phism.  11. More general complements and “ all-or-nothing” transversality In summar y the idea that we hav e applied a b ove is that first we hav e build a DG-mo dule mo del of C ∗ ( W r ∪ e ∈ E X e ) of the form of a total cofibre s − n T otCof ( γ 7→ # C ∗ ( X γ )) = s − n ⊕ γ ∈ Γ y γ # C ∗ ( X γ ) . This only requires a mixture o f Lefsc hetz duality and a genera l Mayer-Vietoris prin- ciple. The disadv antage of this mo del, which works for any system of subp olyhedra { X e ֒ → W } e ∈ E is that this mo del has no clea r CDGA structur e, partly b ecause there is no such algebra structure on the duals # C ∗ ( X γ ). In the ca se of the c o nfiguration space there was another mo del which (non- equiv ar ia ntly at lea st) is is omorphic to s − n ⊕ γ ∈ Γ 0 Y e ∈ γ g e C ∗ ( X γ ) which admits a clearer alg e bra structure. T o build this mo del we have applied Poincar´ e duality at the co chain level for eac h o f the submanifold X γ , s − n y γ # C ∗ ( X γ ) ≃ Q e ∈ γ g e C ∗ ( X γ ). F o r this to make sense we nee d first each of the X γ to b e a c lo sed manifold, but also for a ll these Poincar´ e dua lities a t v a rious fo rmal dimensions to fit to gether to recov er the Lefschetz duality fo r C ∗ ( W r ∪ e ∈ E X e ) we nee de d some sor t of transversality . In a sense it is exaclty to recov er this transversality that we had to restrict to a s ubset Γ 0 ⊂ Γ defined at the b eginning of the pro of of Lemma 10.1 1. In fact this a ppr oach can b e applied to mor e g e ne r al space than configura tion spaces. Actually the ma in points tha t we her e used is the fact that we had an ori- ent ed manifold W tog e ther with a system of clo sed s ubmanifolds X • := { X e ֒ → W } e ∈ E such that the families of intersections {∩ e ∈ γ X e } γ ∈ Γ satisfies a certain “all-or -nothing” transversality condition that we now explain. MODELS F OR CONFIGURA TION S P ACES 23 In the c ase wher e X • is a t otal tr ansverse system of submanifolds, by whic h we mean that for any disjoint γ 1 , γ 2 ⊂ E the submanifolds ∩ e 1 ∈ γ 1 X e 1 and ∩ e 2 ∈ γ 2 X e 2 in- tersects transversally , then us ing co chain-level Poincar´ e duality gives another model of C ∗ ( W r ∪ e ∈ E X e ) o f the form ( ⊕ γ ∈ Γ g γ .C ∗ ( X γ ) , D ) where deg ( g γ ) = co dim( X γ ), a nd ther e is a natura l CDGA structure o n this when we think to g γ as Q e ∈ γ g e . In other w ords in the case of a total transv ers e system we ca n tak e Γ 0 = Γ. In the case of the configura tion s pa ce we have E = { ( i, j ) : 1 ≤ i < j ≤ k } and the familly of diag onals X • := { X ij ֒ → M k } ( i,j ) ∈ E is certa inly no t totally transverse, except when k ≤ 2. But it has ano ther prop erty whic h we call al l-or- nothing tr ansverse . By this we mean that for any γ 1 , γ 2 ⊂ E the submanifolds ∩ e 1 ∈ γ 1 X e 1 and ∩ e 2 ∈ γ 2 X e 2 either intersects tr ansversally or one o f them is included in the other. This is the case of the system of diagona ls in M k . Using that it is then alwa ys p ossible to find a subset Γ 0 ⊂ Γ such that C ∗ ( W r ∪ e ∈ E X e ) has a mo del of the for m ( ⊕ γ ∈ Γ 0 u γ .C ∗ ( X γ ) , D ) where u γ = codim( X γ ), a nd again this c o mes with a natural CDGA str ucture on this. Actually the subset Γ 0 ⊂ Γ is characterized b y the fact that if γ ∈ Γ 0 and γ ′ ⊂ γ then γ ′ ∈ Γ 0 and, for e 6∈ γ , we hav e γ ∪ { e } ∈ Γ 0 if and o nly if X ′ γ ∪ { e } 6 = X γ . W e do no t claim that it is a CDGA mo del in general, and finding s uitable conditions for this to b e true is certainly an in teresting but difficult problem. 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P.L.: Universit ´ e Ca tholique de Louv ain, Institut Ma th ´ ema tique, 2, chem in du Cy- clotron, B-13 4 8 Louv ain-la-Neuve, BELGIUM E-mail addr ess : pascal.lamb rechts@uclouvain.be D.S.: University of Regina. Saska tchew an. CANADA. E-mail addr ess : Donald.Stan ley@uregina.ca