Hochschild cohomology of Sullivan algebras and mapping spaces between manifolds
aa r X i v : . [ m a t h . A T ] A ug HOCHSCHILD COHOMOLOGY OF SULLIVANALGEBRAS AND MAPPING SPACES BETWEENMANIFOLDS
J.-B. GATSINZI
Abstract.
Let e : N n → M m be an embedding into a compactmanifold M . We study the relationship between the homology ofthe free loop space LM on M and of the space L N M of loopsof M based in N and define a shriek map e ! : H ∗ ( LM, Q ) → H ∗ ( L N M, Q ) using Hochschild cohomology and study its proper-ties. We also extend a result of F´elix on the injectivity of the in-duced map aut M → map( N, M ; f ) on rational homotopy groupswhen M and N have the same dimension and f : N → M is a mapof non zero degree. Introduction
All spaces are assumed to be simply connected and (co)homologycoefficients are taken in the field Q of rationals. If M is a compactoriented manifold of dimension m and LM = map( S , M ) the space offree loops in M , then there is an intersection product µ : H p + m ( LM ) ⊗ H q + m ( LM ) → H p + q + m ( LM )which induces a graded multiplication on H ∗ ( LM ) = H ∗ + m ( LM ) turn-ing into a graded algebra [3]. Consider the embedding e : N → M of asubmanifold of degree n . Construct the pullback L N M ˜ e / / ˜ p (cid:15) (cid:15) LM p (cid:15) (cid:15) N e / / M, where p is the evaluation of a loop at 1 ∈ S . There is also an intersec-tion product on H ∗ ( L N M ) = H ∗ + n ( L N M ), turning it into commutative Mathematics Subject Classification.
Primary 55P62; Secondary 54C35.
Key words and phrases.
Loop space homology, Poincar´e duality, Hochschildcohomology.A partial support from the IMU-Simons Africa Fellowship is acknowledged. graded algebra [15].We consider a morphism f : ( A, d ) → ( B, d ) of commutative dif-ferential graded algebras which models the embedding e , where ( A, d )and (
B, d ) are Poincar´e duality algebras [5]. We show that there is an A -linear shriek map f ! : ( B, d ) → ( A, d ) of degree m − n . We considerinduced maps HH ∗ ( f ) : HH ∗ ( A ; A ) → HH ∗ ( A ; B ) and HH ∗ ( f ! ) : HH ∗ ( A ; B ) → HH ∗ ( A ; A ) in Hochschild cohomology. Moreover weobtain the following. Theorem 1.
The composition map HH ∗ ( f ! ) ◦ HH ∗ ( f ) : HH ∗ ( A ; A ) → HH ∗ ( A ; A ) is the multiplication by the Poincar´e dual of the fundamental class of N in M . Theorem 2.
Let g : N m → M m be a map between manifolds of samedimension m such deg f = 0 and f : ( A, d ) → ( B, d ) a cdga model of g . Then HH ∗ ( A ; A ) → HH ∗ ( A ; B ) is injective. The above result suggests that H (˜ g ) : H ∗ ( L N M ) → H ∗ ( LM ) is an in-jective algebra homomorphism, where ˜ g : L N M → LM is the pullbackof g : N → M along the fibration p : LM → M defined by p ( γ ) = γ (0) . The paper is organized as follows: In Section 2 we define a shriekmap f ! : ( B, d ) → ( A, d ) and prove Theorem 1. In Section 3, we recall aresolution to compute HH ∗ ( C ∗ ( M ) , C ∗ ( N )) and in Section 4 we proveTheorem 2. 2. A shriek map
We first recall some facts in Rational Homotopy Theory. We makeuse of Sullivan models for which the standard reference is [6]. All vec-tor spaces are over the ground field Q . A differential graded algebra( A, d ) is a direct sum of vector spaces A p , that is, A = ⊕ p ≥ A p togetherwith a graded multiplication µ : A p ⊗ A q → A p + q which is associative.An element a ∈ A p is called homogeneous of degree | a | = p . Moreoverthere is a differential d : A p → A p +1 which an algebra derivation, thatis, d ( ab ) = ( da ) b + ( − | a | a ( db ) and satisfies d = 0.The algebra A is commutative if ab = ( − | a || b | ba . If ( A, d ) is a com-mutative differential graded algebra (cdga for short), then H ∗ ( A, d ) is
APPING SPACES BETWEEN MANIFOLDS 3 graded commutative. A morphism f : ( A, d ) → ( B, d ) of cdga’s iscalled a quasi-isomorphism if H ∗ ( f ) is an isomorphism. A cdga ( A, d )is called simply connected if H ( A ) = Q and H ( A ) = 0.A commutative graded algebra A is free if it is of the form ∧ V = S ( V even ) ⊗ E ( V odd ), where V even = ⊕ i ≥ V i and V odd = ⊕ i ≥ V i +1 .A Sullivan algebra is a cdga ( ∧ V, d ), where V = ⊕ i ≥ V i admits a ho-mogeneous basis { x i } i ∈ I indexed by a well ordered set I such dx i ∈∧ ( { x i } ) i A, d ).To a simply connected topological space X of finite type, Sullivanassociates in a functorial way a cdga A P L ( X ) of piecewise linear formson X such H ∗ ( A P L ( X )) ∼ = H ∗ ( X, Q ) [16]. A Sullivan model of X is a Sullivan model of A P L ( X ). Moreover any cdga ( A, d ) is called acdga-model of X if there is a sequence of quasi-isomorphisms( A, d ) → ( A , d ) ← . . . → ( A n − , d ) ← A P L ( X ) . We state here the fundamental result of Sullivan algebras. Proposition 3. If ( A, d ) is a simply connected cdga then there is aminimal Sullivan algebra ( ∧ V, d ) together with a quasi-isomorphism ( ∧ V, d ) → ( A, d ) . Moreover ( ∧ V, d ) is unique up to isomorphism. It iscalled the minimal Sullivan model of ( A, d ) [6, § .Definition . Let X be a simply connected space. A minimal Sullivanmodel ( ∧ V, d ) of X is the minimal Sullivan model of A P L ( X ). It iscalled formal if there is a quasi-isomorphism ( ∧ V, d ) → H ∗ ( ∧ V, d ). Inthis case X is called a formal space. Formal spaces include spheres,compact Lie groups and complex projective spaces. Definition . A commutative differential graded algebra ( A, d ) is aPoincar´e algebra of formal dimension n if A is connected and thereis a linear map ǫ : A n → Q such that(1) ǫ ( dA n − ) = 0,(2) the bilinear form b : A k ⊗ A n − k → Q , defined by b ( x ⊗ y ) = ǫ ( ab )is non degenerate. Remark . If A is of finite type, then A i = 0 for i > n and A is finitedimensional. Moreover if { a , . . . , a k } is a homogeneous basis of A ,then there is a dual homogeneous basis { a ∗ j } such that ǫ ( a i a ∗ j ) = δ ij .We denote by a the dual of a in A = Hom( A, Q ). In particular ω A = J.-B. GATSINZI ǫ ∈ ( A ) ∼ = A is the fundamental class of A . Moreover there is anisomorphism of A -modules π A : A → A defined by π A ( a )( x ) = b ( ax ) . If ( ∧ V, d ) is the minimal Sullivan model of a simply connected space X , where H ∗ ( X, Q ) satisfies Poincar´e duality, then ( ∧ V, d ) is quasi-isomorphic to a Poincar´e duality algebra ( A, d ) [13]. In particular, asimply connected smooth manifold M of dimension m has a cdga-model( A, d ) which satisfies Poincar´e duality in dimension m .Let f : ( A, d ) → ( B, d ) be a map between cdga’s with Poincar´eduality in dimensions m and n respectively. We can now relate isomor-phisms π A : A ≃ → A and π B : B ≃ → B . Proposition 7. If f is surjective, then there exists a morphism of A -modules f ! : B → A making the following diagram commutative. B ≃ π B (cid:15) (cid:15) f ! / / A ∼ = π A (cid:15) (cid:15) B f / / A Proof. Let 1 ∈ B , then π B (1) = ω B , where ω B is a cocycle whichrepresents the fundamental class [ ω B ] ∈ H n ( B ). As π A is bijective,there exists α ∈ A such that π A ( α ) = f ( ω B ). As f is surjective, thengiven b ∈ B , there exists a ∈ A such that b = f ( a ). Recall that B is an A -module through the action induced by f , hence b = f ( a )1 = a ∗ f ! ( b ) = aα . In particular f ! f ( a ) = aα .We show that the above diagram commutes. Let b ∈ B and a ∈ A suchthat b = f ( a ). On one hand(1) f ( π B ( b )) = f ( π B ( b × f ( bω B ) , whereas(2) π A ( f ! ( b )) = π A ( aα ) = aπ A ( α ) = af ( ω B ) . Let x ∈ A . Then(3) f ( bω B )( x ) = ( bω B )( f ( x )) = ω B ( bf ( x )) , and(4) ( af ( ω B ))( x ) = ( f ( ω B ))( ax ) = ω B ( f ( ax ))= ω B ( f ( a ) f ( x )) = ω B ( bf ( x )) . Hence f ( bω B ) = af ( ω B ) and the diagram commutes. APPING SPACES BETWEEN MANIFOLDS 5 Finally we show that f ! is a morphism of A -modules. If x ∈ A and b ∈ B , then f ! ( x ∗ b ) = f ! ( f ( x ) b ) = f ! ( f ( xa )) = ( xa ) α = xf ! ( b ) . In particular f ! ( b ) = f ! ( b × 1) = a ∗ f ! (1). (cid:3) Remark . If ω B is a cocycle representing the fundamental class of( B, d ) and f is surjective, then there exists x ∈ A such that f ( x ) = ω B .Then f ( ω B ) = x = π A ( x ∗ ), where x ∗ is the dual of x under a choiceof a basis { a i } of A and its dual { a ∗ j } (see Remark 6). If dx = 0, then[ x ] ∈ H ∗ ( A ) = 0 and [ x ∗ ] ∈ H m − n ( A ) is non zero. Example . Consider the inclusion i : C P n → C P n + k . As complexprojective spaces are formal, a cdga model of the inclusion is f : ∧ x / ( x n + k +12 ) → ∧ y / ( y n +12 ) , where f ( x ) = y . Then f ! is defined by f ! (1) = x k . Hence f ! ( y i ) = x k + i , for 0 ≤ i ≤ n . 3. Hochschild cohomology If ( A, d ) is a graded differential algebra and ( M, d ) a graded A -bimodule, then the Hochschild cohomology of A with coefficients in M is defined by HH ∗ ( A ; M ) = Ext A e ( A, M ), where A e = A ⊗ A opp .Let A = ( ∧ V, d ) be the minimal Sullivan model of a simply connectedspace X . Then(5) P = ( ∧ V ⊗ ∧ V ⊗ ∧ ¯ V , ˜ D ) → ( ∧ V, d )is a semi-free resolution of ∧ V as a ∧ V ⊗∧ V -module, where ¯ V = sV [5].Moreover, the pushout( ∧ V ⊗ ∧ V, d ⊗ d ⊗ µ (cid:15) (cid:15) / / / / ( ∧ V ⊗ ∧ V ⊗ ∧ ¯ V , ˜ D ) (cid:15) (cid:15) ( ∧ V, d ) / / / / ( ∧ V ⊗ ∧ ¯ V , D )yields a Sullivan model ( ∧ V ⊗ ∧ ¯ V , D ) of the free loop space on X [17].The differential is given by Dv = dv for v ∈ V and D ¯ v = − Sdv , where S is the unique derivation on ∧ V ⊗ ∧ ¯ V defined by Sv = ¯ v and S ¯ v = 0.Hence if ( M, d ) is a ∧ V -differential module, then the Hochschild cochains CH ( A ; M ) are given by(6) CH ∗ ( A ; M ) = (Hom ∧ V ⊗∧ V ( ∧ V ⊗ ∧ V ⊗ ∧ ¯ V , M ) , D ) ∼ = (Hom ∧ V ( ∧ V ⊗ ∧ ¯ V , M ) , D ) . J.-B. GATSINZI As the differential of D on ∧ V ⊗ ∧ ¯ V satisfies D ( ∧ V ⊗ ∧ n ¯ V ) ⊂ ∧ V ⊗ ∧ n ¯ V , one gets a Hodge type decomposition HH ∗ ( A ; M ) = ⊕ i ≥ HH ∗ ( i ) ( A ; M ) , where HH ∗ ( i ) ( A ; M ) = H ∗ (Hom ∧ V ( ∧ V ⊗ ∧ i ¯ V , ∧ V ) , D ). Moreover, if L = s − Der ∧ V , then the symmetric algebra ( ∧ A L, d ) is quasi-isomorphicto the Hochschild cochain complex (Hom ∧ V ( ∧ V ⊗ ∧ ¯ V , ∧ V ) , D ) [9]. If( ∧ V, d ) the minimal Sullivan model of a simply connected smooth com-pact and oriented manifold M of dimension m , then there is an iso-morphism of BV-algebras H ∗ ( LM ) ∼ = HH ∗ ( ∧ V ; ∧ V ) [4, 8, 7].Let M be a smooth compact, oriented and simply connected manifoldof dimension m . For submanifolds N and N ′ , we denote by L N ′ N M the space of paths in M starting in N and ending in N ′ , and L NN M issimply written L N M . Let N , N and N be submanifolds of M . Whencoefficients are rationals (or in Z /n Z ) Sullivan showed that there is anintersection product µ : H p + d ( L N N M ) ⊗ H q + d ( L N N M ) → H p + q + d ( L N N M )where d = dim N [15]. In particular if N = N = N = N , one gets agraded commutative algebra structure on H ∗ ( L N M, Q ) = H ∗ + d ( L N M, Q ).Let e : N n ֒ → M m be an embedding where N is simply connectedand f : ( A, d ) → ( B, d ) a cdga model of e , where both ( A, d ) and( B, d ) satisfy Poincar´e duality. Assume that f is surjective and let[ y ] ∈ H n ( B ) be the fundamental class. Let x ∈ A such that f ( x ) = y .We will assume that x is a cocycle and consider [ x ] ∈ H n ( A, d ). Theorem 10. Under the above hypotheses, the composition HH ∗ ( A ; A ) HH ∗ ( f ) / / HH ∗ ( A ; B ) HH ∗ ( f ! ) / / HH ∗ ( A ; A ) is the multiplication with the Poincar´e dual [ x ∗ ] ∈ H m − n ( A, d ) of [ x ] .Proof. We consider a minimal Sullivan model φ : ( ∧ V, d ) → ( A, d ). ByEq. (6), HH ∗ ( A ; A ) is obtained as the cohomology of the complexHom ∧ V ⊗∧ V ( ∧ V ⊗ ∧ V ⊗ ∧ ¯ V , ∧ V ) ∼ = Hom ∧ V ( ∧ V ⊗ ∧ ¯ V , ∧ V ) ≃ Hom ∧ V ( ∧ V ⊗ ∧ ¯ V , A ) . If γ ∈ Hom ∧ V ( ∧ V ⊗ ∧ ¯ V , A ), then( CH ( f ! ) ◦ CH ( f ))( γ )( x ) = ( f ! ◦ f )( γ )( x ) = αγ ( x ) , APPING SPACES BETWEEN MANIFOLDS 7 where α = x ∗ , by Remark 8. Therefore, if γ is a cocycle, then HH ∗ ( f ! ) ◦ HH ∗ ( f ) = [ x ∗ ][ γ ]. (cid:3) Example . The hypotheses of Theorem 10 are satisfied if e : N → M is an embedding between formal smooth manifolds where H ∗ ( e )is surjective, for instance the inclusion C P n → C P n + k . Let A = H ∗ ( C P n + k , Q ) = ∧ x / ( x n + k +12 ). The complex to compute HH ∗ ( A ; A )is given by ( A ⊗ ∧ ( z , z n + k ) ) , D ) where subscripts indicate the lowerdegree, and Dz n + k ) = 0 and Dz = ( n + k + 1) x n + k z n + k ) [10]. Herean element x ∈ A n = A − n is assumed to be of lower degree − n . Atchain’s level, the composition CH ∗ ( f ! ) ◦ CH ( f ) : ( A ⊗ ∧ ( z , z n + k ) ) , D ) → ( A ⊗ ∧ ( z , z n + k ) ) , D )is the multiplication by x k .If e : N → M is an embedding between manifolds, then L N M is thepullback of the following diagram(7) L N M ˜ e / / ˜ p (cid:15) (cid:15) LM p (cid:15) (cid:15) N e / / M, where p ( γ ) = γ (0).Assume that π ∗ ( M ) ⊗ Q is finite dimensional and ( ∧ V, d ) is the min-imal Sullivan model of M . Then HH ∗ ( ∧ V ; ∧ V ) is the homology of thecomplex ( ∧ V ⊗ ∧ Z, D ) where Z ≃ s − V [10]. Proposition 12. If f : ( A, d ) → ( B, d ) is a model of e : N → M , then HH ∗ ( C ∗ ( M ); C ∗ ( N )) is computed by the complex ( B ⊗∧ Z, D ) obtainedas the pushout (8) ( A, d ) / / (cid:15) (cid:15) ( A ⊗ ∧ Z, D ) (cid:15) (cid:15) ( B, d ) / / ( B ⊗ ∧ Z, D ) Proof. Let ( ∧ V, d ) is the minimal Sullivan model of M , where V isfinite dimensional. Then H ∗ ( LM ) is the homology of the complex( ∧ V ⊗ ∧ Z, D ) where Z = s − V and the differential D is inducedby δ on (Der ∧ V, δ ) where V ⊂ Der ∧ V . As ( ∧ V, D ) → ( A, d ) isa quasi-isomorphism, then the pushout is a model of the pullback inEq. 7. (cid:3) J.-B. GATSINZI However, it is known whether structure of H ∗ ( L N M ) and H ∗ ( B ⊗∧ Z, D ) are isomorphic as algebras.4. Maps between manifolds of same dimension Let f : ( A, d ) → ( B, d ) be a morphism of graded cochain algebras.An f -derivation of degree n is a linear map θ : A ∗ → B ∗− n suchthat θ ( xy ) = θ ( x ) f ( y ) + ( − n | x | f ( x ) θ ( y ). We denote by Der n ( A, B ; f )the vector space of all f -derivations of degree n and Der( A, B ; f ) = ⊕ n Der n ( A, B ; f ). Define a differential δ on Der( A, B ; f ) by δθ = d B θ − ( − | θ | θd A . If A = B , then we simply write Der A for Der( A, A ; 1 A ).The graded vector space Der A is endowed with the commutator bracketturning it into a graded differential Lie algebra. There is an action of A on Der A , defined by ( aθ )( x ) = aθ ( x ), making (Der A, δ ) a differentialgraded module over A .Let M and N be compact and oriented manifolds of dimension n and g : N → M a smooth map such that deg g = 0. Consider aPoincar´e duality model f : ( A, d ) → ( B, d ) of g . Then f is injectiveand B = f ( A ) ⊕ Z , where dZ ⊆ Z and f ( A ) .Z [5]. Therefore Z is an A -submodule. Moreover the projection p : B = f ( A ) ⊕ Z → A is amorphism of A -modules. Theorem 13 ([5], Theorem 2) . Consider a surjective Sullivan model φ : ( ∧ V, D ) → ( A, d ) . Then (9) f ∗ : (Der( ∧ V, A ; φ ) , δ ) → (Der( ∧ V, B ; f ◦ φ ) , δ ) induces an injective map in homology. This can be interpreted in terms of rational homotopy groups offunction spaces. Let g : X → Y be a continuous map between CWcomplexes where Y is finite and X of finite type and φ : ( ∧ Z, d ) → ( B, d ) a Sullivan model of g . Consider map( X, Y ; g ) be the space ofcontinuous mappings from X to Y which are homotopic to f . There isa natural isomorphism [1, 2, 14] π n (map( X, Y ; g )) ⊗ Q ∼ = H n (Der( ∧ V, B ; φ ) , δ ) , n ≥ . Hence if g : N → M is a map between simply connected smoothmanifolds such that deg g = 0, then the map j M : aut M = map( M, M ; 1 M ) → map( N, M ; g )induces an injective map π ∗ ( j M ) ⊗ Q : π ∗ (aut M ) ⊗ Q → π ∗ (map( N, M ; g )) ⊗ Q . APPING SPACES BETWEEN MANIFOLDS 9 Let φ : ( ∧ V, d ) → ( A, d ) be a Sullivan model and ρ = f ◦ φ . We havethe following commutative diagram H ∗ (Der ∧ V, δ ) (cid:15) (cid:15) (cid:15) (cid:15) (cid:31) (cid:127) / / HH ∗ ( A ; A ) (cid:15) (cid:15) H ∗ (Der( ∧ V, B ; ρ ) , δ ) (cid:31) (cid:127) / / HH ∗ ( A ; B ) , where horizontal maps are inclusions [11]. We show that the remainingvertical arrow is injective, which is a generalization of Theorem 13. Theorem 14. Let g : N → M be a smooth map between manifolds and f : ( A, d ) → ( B, d ) a Poincar´e duality model of g . Then the inducedmap HH ∗ ( A ; A ) HH ∗ ( f ) / / HH ∗ ( A ; B ) is injective.Proof. As B = f ( A ) ⊕ Z , then f ( A ) = ρ ( ∧ V ) is a submodule of B viewed as a ∧ V -module and Z is also a ∧ V -submodule of B . ThereforeHom ∧ V ( ∧ V ⊗∧ ¯ V , B ) ∼ = Hom ∧ V ( ∧ V ⊗∧ ¯ V , f ( A )) ⊕ Hom ∧ V ( ∧ V ⊗∧ ¯ V , Z ) . Moreover, the projection p : B = f ( A ) ⊕ Z → f ( A ) ∼ = A is a morphismof ∧ V -modules. It induces a chain map p ∗ : Hom ∧ V ( ∧ V ⊗ ∧ ¯ V , B ) → Hom ∧ V ( ∧ V ⊗ ∧ ¯ V , A )such that p ∗ ◦ f ∗ is the identity. Therefore f ∗ is injective in homology. (cid:3) We can then deduce the following Corollary 15. Under the hypotheses of Theorem 14, there is an injec-tive map H ∗ ( f ) ! : H ∗ ( LM, Q ) → H ∗ ( L N M, Q ) Proof. 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