On The Group Of Self-homotopy Equivalences Of An Elliptic Space
aa r X i v : . [ m a t h . A T ] O c t ON THE GROUP OF SELF-HOMOTOPY EQUIVALENCES OFAN ELLIPTIC SPACE
MAHMOUD BENKHALIFA
Abstract.
Let X be a simply connected rational elliptic space of formal di-mension n and let E ( X ) denote the group of homotopy classes of self-equivalencesof X . If X [ k ] denotes the k th Postikov section of X and X k denotes its k th skeleton, then making use of the models of Sullivan and Quillen we prove that E ( X ) ∼ = E ( X [ n ] ) and if n > m = max (cid:8) k | π k ( X ) = 0 (cid:9) and E ( X ) is finite, then E ( X ) ∼ = E ( X m +1 ). Moreover, in case when X is 2-connected, we show that if π n ( X ) = 0, then the group E ( X ) is infinite. Introduction
A simply connected rational topological space X is elliptic if both H ∗ ( X, Q ) and π ∗ ( X ) are finite dimensional. Let us call n = max { i : H i ( X, Q ) = 0 } the formaldimension of X . It is well known ([3], Theorem 4.2) that a such space satisfies π i ( X ) = 0 , for i ≥ n. Therefore, if X [ k ] denotes the k th Postikov section of X and X k denotes its k th skeleton, on the one hand X coincides with X [2 n − and on the other hand, as itsformal dimension is n , the space X coincides with X n .This paper is directed towards an understanding of the group of homotopy classesof self-equivalences E ( X ), where X is a simply connected rational elliptic space offormal dimension n . As is well known, the homotopy theory of rational spacesis equivalent, by Sullivan’s work, to the homotopy theory of minimal, differential,graded commutative Q -algebras and by Quillen’s work to the homotopy theory ofdifferential, graded Lie Q -algebras. Those algebras provide an effective algebraicsetting to work in, so working algebraically we establish the following theorem whichwill be split in Theorem 3.3 and Theorem 3.4 later on. Theorem 1.
Let X be a simply connected rational elliptic space of formal dimen-sion n . Then E ( X ) = E ( X [2 n − ) ∼ = E ( X [2 n − ) ∼ = . . . ∼ = E ( X [ n ] ) Moreover if n > m = max (cid:8) k | π k ( X ) = 0 (cid:9) and E ( X ) is finite, then E ( X ) ∼ = . . . ∼ = E ( X m +2 ) ∼ = E ( X m +1 ) . The idea of getting some information regarding the (in)finiteness of the groups E ( X ) within the framework of Sullivan model traces back to the results of Arkowitzand Lupton [2] in which they exhibited conditions under which E ( X ) is finite or Mathematics Subject Classification.
Key words and phrases.
Elliptic spaces, Group of homotopy self-equivalences, Quillen model,Sullivan model, Whitehead exact sequence. infinite, where X is a rational space having a 2-stage Postnikov-like decomposi-tion (for example, rationalizations of homogeneous spaces). In the same spirit weestablish the following result which will be Theorem 3.5 later on. Theorem 2.
Let X be a -connected rational elliptic space of formal dimension n .If π n ( X ) = 0 , then the group E ( X ) is infinite. In [15], Costoya and Viruel proved the remarkable result that every finite group G occurs as G = E ( X ) for some elliptic rational space X having formal dimension n = 208+80 | V | , where V is a certain finite graph associated with G and | V | denotesthe order of V . The space X is constructed such that π k ( X ) = 0 for all k ≥ G can be realised by a rational space X whose formal dimension does not depend onthe order of G . Precisely, we can ameliorate Costoya and Viruel theorem by showingthat G = E ( X ) for some rational space X having formal dimension n = 120.For a space X , let [ X, X ] denote the monoid of homotopy classes of self-maps ofthe space X and let A k ( X ) = n f ∈ [ X, X ] | π i ( f ) : π i ( X ) ∼ = −→ π i ( X ) for any i ≤ k } In [14], Choi and Lee introduced the concept of the self-closeness number definedas follows N E ( X ) = min n k | A k ( X ) = E ( X ) o Notice that the study of this numerical homotopy invariant by means of Sullivanmodel in algebraic setting is an interesting problem (see for instance [20]). From themain result in this paper we establish the following result which will be Theorem3.7 later on.
Theorem 3. If X is a simply connected rational elliptic space of formal dimension n , then N E ( X ) ≤ n Remark . Theorem 3 is a weaker version of ([14], Theorem 2) in which the sameresult is proved, using topological arguments, for a CW-complex X of dimension n . Our proof is based on the analysis of the Sullivan model of X .The paper is organized as follows. In section 2, we recall the basic properties ofthe Quillen model and Sullivan model in rational homotopy theory, the Whiteheadexact sequences as well as the important properties of elliptic spaces. Then weformulate and prove the main theorems in the algebraic setting. In section 3, amere transcription of the above results in the topological context is given and someexamples are provided illustrating our results.2. main results Quillen model and Sullivan model in rational homotopy theory.
Webriefly recall Quillen’s differential graded Lie algebra and Sullivan’s commutativedifferential algebra frameworks for rational homotopy theory. All the materials canbe founded [3, 18].If X is a simply connected rational CW complex of finite type, then there existsa free commutative cochain algebra (Λ V, ∂ ) called the Sullivan model of X , unique up to isomorphism, which determines completely the homotopy type of the space X . Moreover Sullivan model recovers homotopy data via the identifications :Hom (cid:0) π ∗ ( X ) , Q (cid:1) ∼ = V ∗ , H ∗ ( X ; Q ) ∼ = H ∗ (Λ V, ∂ ) and E ( X ) ∼ = aut(Λ V, ∂ ) / ≃ , where aut(Λ V, ∂ ) / ≃ is the group of homotopy cochain self-equivalences of (Λ V, ∂ )modulo the relation of homotopy between free commutative cochain algebras (see[18]). We write E (Λ V ) = aut(Λ V, ∂ ) / ≃ for this group.Dually, if X is a simply connected rational CW complex of finite type, thenthere exists a differential graded Lie algebra ( L ( W ) , δ ) called the Quillen model of X , unique up to isomorphism, which determines completely the homotopy type ofthe space X . The Quillen model recovers homotopy data via the identifications : π ∗ ( X ) ∼ = H ∗− ( L ( W )) and H ∗ ( X, Q ) ∼ = W ∗ . Quillen’s theory directly implies an identification E ( X ) ∼ = aut( L ( W )) / ≃ , where the latter is the group of homotopy differential graded Lie self-equivalencesof ( L ( W ) , δ ) modulo the relation of homotopy between differential graded Lie au-tomorphisms (see [18, pp.425-6]). We write E ( L ( W )) = aut( L ( W ) , ∂ ) / ≃ for this group. Definition 2.1 ([6], Definition 2.6) . Given a simply connected free commutativecochain algebra (Λ( V q ⊕ V ≤ n ) , ∂ ), where q > n and let b q : V q → H q +1 (Λ( V ≤ n ))be the linear map defined as follows b q ( v ) = [ ∂ ( v )] , v ∈ V q . (1)Here [ ∂ ( v )] denotes the cohomology class of ∂ ( v ) ∈ (Λ V ≤ n ) q +1 .We define D qn to be the subgroup of aut( V q ) × E (Λ V ≤ n ) consisting of the pairs( ξ, [ α ]) making the following diagram commutes V q ✲ V qb q b q ❄ ❄ ξH q +1 ( α ) H q +1 (Λ V ≤ n ) ✲ H q +1 (Λ V ≤ n ) (2) Theorem 2.2 ([6], Theorem 1.1) . There exists a short exact sequence of groups
Hom (cid:0) V q , H q (Λ( V ≤ n )) (cid:1) E (Λ( V q ⊕ V ≤ n )) Ψ ։ D qn (3) where Ψ([ α ]) = ( e α q , [ α n ]) . Here e α q : V q → V q is the isomorphism induced by α onthe indecomposables and α n is the restriction of α to Λ V ≤ n . Corollary 2.3.
Assume that the linear map b q is an isomorphism, then D qn ∼ = E (Λ V ≤ n ) MAHMOUD BENKHALIFA
Proof. As b q is an isomorphism, then from the commutative diagram (2) we deducethat ξ = ( b q ) − ◦ H q +1 ( α ) ◦ b q . Therefore the map E (Λ V ≤ n ) → D qn , [ α ] (cid:16) ( b q ) − ◦ H q +1 ( α ) ◦ b q , [ α ] (cid:17) is an isomorphism. (cid:3) Definition 2.4 ([7], Definition 2.1) . Given a simply connected free differentialgraded Lie ( L ( W q ⊕ W ≤ k ) , δ ) where q > k and let b q : W q → H q − ( L ( W ≤ k )) be thelinear map defined as follows b q ( v ) = [ δ ( v )] , v ∈ W q (4)Here [ δ ( v )] denotes the homology class of δ ( v ) ∈ L q − ( W ≤ k ).We define R qk to be subgroup of aut( W q ) × E ( L ( W ≤ k ) consisting of the pairs( ξ, [ α ]) making the following diagram commutes W q ✲ W qb q b q ❄ ❄ ξH q − ( α ) H q − ( L ( W ≤ k )) ✲ H q − ( L ( W ≤ k )) (5) Theorem 2.5 ([7], Theorem 2.6) . There exists a short exact sequence of groups
Hom (cid:0) W q , H q ( L ( W ≤ k )) (cid:1) E ( L ( W q ⊕ W ≤ k )) λ ։ R qk (6) where λ ([ α ]) = ( e α q , [ α k ]) . Here e α q : W q → W q is the isomorphism induced by α onthe indecomposables and α k is the restriction of α to L ( W ≤ k ) Corollary 2.6.
Assume that the linear map b q is an isomorphism, then R qk ∼ = E ( L ( W ≤ k )) Proof. As b q is an isomorphism, then from the commutative diagram (5) we deducethat ξ = ( b q ) − ◦ H q − ( α ) ◦ b q . Therefore the map E ( L ( W ≤ k )) → R qk , [ α ] (cid:16) ( b q ) − ◦ H q − ( α ) ◦ b q , [ α ] (cid:17) is an isomorphism. (cid:3) Whitehead exact sequences in rational homotopy theory.
To everyfree differential graded Lie algebra ( L ( W ) , δ ) such that W = 0, we can assign (see[4, 5, 7, 11]) the following long exact sequence · · · → W n +1 b n +1 −→ H n ( L ( W ≤ n − )) → H n ( L ( W )) → W n b n +1 −→ · · · (7)called the Whitehead exact sequence of ( L ( W ) , δ ), where b ∗ is the graded linearmap defined in (4). Hence if X is a 2-connected rational space of finite type and if( L ( W ) , ∂ ) is its Quillen’s model, then the properties of this model imply π n ( X ) ∼ = H n − ( L ( W )) , H n ( X, Q ) ∼ = W n − , π n ( X n − ) ∼ = H n − ( L ( W ≤ n − )) Here X n for the n th skeleton of X . Therefore the Whitehead exact sequence of thismodel can be written as · · · → H n +1 ( X ) → π n ( X n − ) → π n ( X ) → H n ( X ) → · · · (8)Likewise, let (Λ V, ∂ ) be a simply connected free commutative cochain algebra.In [8, 9, 10, 13, ? ], it is shown that with (Λ V, ∂ ) we can associate the following longexact sequence · · · → V k b k −→ H k +1 (Λ V ≤ k − ) → H k +1 (Λ V ) → V k +1 b k +1 −→ · · · (9)called the Whitehead exact sequence of (Λ V, ∂ ), where b ∗ is the graded linear mapdefined in (1). Thus, if X is a simply connected rational space of finite type and(Λ( V ) , ∂ ) is its Sullivan minimal model, then by virtue of the properties of thismodel we obtain the following identifications H k ( X, Q ) ∼ = H n (Λ V ) , H k +1 ( X [ k ] , Q ) ∼ = H k +1 (Λ V ≤ k ) , V k ∼ = Hom( π k ( X ) , Q )Here X [ k ] for the k th Postikov section of X . Therefore the Whitehead exact se-quence of this model can be written as · · · → Hom( π k ( X ) , Q ) → H k +1 ( X [ k ] ) → H k +1 ( X ) → Hom( π k +1 ( X ) , Q ) → · · · (10) Theorem 2.7. If X is a -connected rational space of finite type, then H k +1 ( X [ k ] , Q ) ∼ = Hom( π k ( X k − ) , Q ) , k ≥ Proof.
Applying the exact functor Hom( ., Q ) to the exact sequence (8) we obtain · · · ← H n +1 ( X, Q ) ← Hom( π n ( X n − ) , Q ) ← Hom( π n ( X ) , Q ) ← H n ( X, Q ) ← · · · (12)Taking into account that • All groups involved are vector spaces of finite dimensions • The two maps H n ( X, Q ) → Hom( π n ( X ) , Q ) appearing in (10) and (12) arethe same morphism • Hom( H ∗ ( X, Q ) , Q ) = H ∗ ( X, Q )and by comparing the sequences (10), (12) we get (11). (cid:3) Elliptic algebras.
Recall that (see [3, 18]) a simply connected free differentialgraded commutative algebra (Λ
V, ∂ ) is called elliptic if both H ∗ (Λ V ) and V ∗ arefinite dimensional. Let us call n = max { i : H i (Λ V ) = 0 } the formal dimensionof (Λ V, ∂ ). The following theorem mentions some important properties of ellipticalgebras.
Theorem 2.8. ( [3] , Theorem 7 . . ). Suppose (Λ V, ∂ ) is simply connected and el-liptic of formal dimension n . Then (1) dim V odd ≥ dim V even . (2) If { x j } is a basis of V odd and { y j } is a basis of V even , then n = X | x j | − X ( | y j | − . (3) P | y j | ≤ n and P | x j | ≤ n − . (4) V i = 0 , for i ≥ n. From the property (3) we can derive
MAHMOUD BENKHALIFA
Corollary 2.9.
Suppose (Λ V, ∂ ) is simply connected and elliptic of formal dimen-sion n . Then V i = 0 , for i > n and i even. Lemma 2.10.
Suppose (Λ V, ∂ ) is simply connected and elliptic of formal dimension n . For every k such that k > n we have H k +1 (Λ V ≤ k − ) = 0 Proof.
From the long exact sequence of cohomology associated to the inclusion(Λ V ≤ k − , ∂ ) ⊆ (Λ V, ∂ ) we get · · · → H k (cid:16) Λ V / Λ V ≤ k − (cid:17) → H k +1 (Λ V ≤ k − ) → H k +1 (Λ V ) → · · · But H k (cid:16) Λ V / Λ V ≤ k − (cid:17) ∼ = V k and by Corollary 2.9 we have V k = 0 for 2 k > n .Therefore the map H k +1 (Λ V ≤ k − ) → H k +1 (Λ V ) is injective. As (Λ V, ∂ ) hasformal dimension n , it follows that H k +1 (Λ V ) = 0 for 2 k ≥ n , implying that H k +1 (Λ V ≤ k − ) = 0 . (cid:3) Using the exact sequence (9), Corollary 2.9 and Lemma 2.10, we derive thefollowing result
Corollary 2.11.
Suppose (Λ V, ∂ ) is simply connected and elliptic of formal dimen-sion n . Then the linear map b i : V i −→ H i +1 (Λ V ≤ i − ) is • An isomorphism for i > n and i odd. • Nil for for i > n and i even. Theorem 2.12.
Suppose (Λ V, ∂ ) is simply connected and elliptic of formal dimen-sion n . Then E (Λ V ) ∼ = E (Λ V ≤ n ) Proof.
First note that since (Λ
V, ∂ ) is elliptic of formal dimension n , by the property(3) of Theorem 2.8, we derive that E (Λ V ) = E (Λ V ≤ n − ) . Next using Theorem 2.2we obtain the following short exact sequenceHom (cid:0) V n − , H n − (Λ V ≤ n − ) (cid:1) E (Λ V ≤ n − ) ։ D n − n − (13)But V n − = 0 and b n − is an isomorphism, so using Corollary 2.3 and Lemma2.10, the sequence (13) implies E (Λ( V ≤ n − )) ∼ = E (Λ( V ≤ n − )Again using theorem 2.2 we obtain the following short exact sequenceHom (cid:0) V n − , H n − (Λ V ≤ n − ) (cid:1) E (Λ V ≤ n − ) ։ D n − n − but V n − = 0 and b n − is an isomorphism, so using Corollary 2.3 and Lemma2.10, the sequence (13) implies E (Λ( V ≤ n − )) ∼ = E (Λ( V ≤ n − )Continuing this process by using the same arguments, we end up with the followingformula E (Λ( V ≤ n − )) ∼ = E (Λ( V ≤ n − ) ∼ = . . . ∼ = E (Λ( V ≤ n ) , if n is oddand E (Λ( V ≤ n − )) ∼ = E (Λ( V ≤ n − ) ∼ = . . . ∼ = E (Λ( V ≤ n +1 ) , if n is even. In the case when n is even, according to Corollary 2.3 and Lemma 2.10, the sequence(13) implies Hom (cid:0) V n +1 , H n +1 (Λ V ≤ n ) (cid:1) E (Λ V ≤ n +1 ) ։ E (Λ V ≤ n )Finally, Lemma 2.10 assures that H n +1 (Λ V ≤ n ) = 0. Hence E (Λ V ≤ n +1 ) ∼ = E (Λ V ≤ n ) (cid:3) Proposition 2.13.
Let (Λ V, ∂ ) be a simply connected free differential graded alge-bra. If the group E (Λ V ≤ n ) is finite, then the linear map b n is injective.Proof. First Theorem 2.2 implies thatHom (cid:0) V n , H n (Λ V ≤ n − ) (cid:1) E (Λ V ≤ n ) ։ D nn − (14)Assume that b n is not injective and let v = 0 ∈ V n such that b n ( v ) = 0. Choose { v, v , . . . , v k } as a basis of V n and define ξ a ( v ) = av , a = 0 ∈ Q , ξ a ( v i ) = v i Clearly the pair ( ξ a , [ id ]) ∈ aut( V n ) × E (Λ V ≤ n − ) for every a = 0 ∈ Q and makesfollowing diagram commute V n ✲ V nb n b n ❄ ❄ ξ a id H n +1 (Λ V ≤ n − ) ✲ H n +1 (Λ V ≤ n − )Therefore ( ξ a , [ id ]) ∈ D nn − for every a = 0 ∈ Q implying that the group D nn − isinfinite. Consequently, the group E (Λ V ≤ n ) is also infinite according to the exactsequence (14) (cid:3) Topological applications
Definition 3.1.
A simply connected rational space X is called elliptic if its Sullivanmodel is elliptic.In this case the formal dimension of X is defined as the formal dimension of itsSullivan model. Remark . By virtue of Theorem 2.8 we conclude that if X is a simply connectedrational elliptic space of formal dimension n , then its Quillen model can be writtenas ( L ( W ≤ n − ) , δ ) and its Sullivan model as (Λ V ≤ n − , ∂ ).A mere transcription of the Theorem 2.12 in the topological context, using theproperties of the Sullivan model, implies the following theorem. Theorem 3.3.
Let X be a rational elliptic space of formal dimension n . Then E ( X ) = E ( X [2 n − ) ∼ = E ( X [2 n − ) ∼ = . . . ∼ = E ( X [ n ] ) Here X [ k ] denotes the k th Postnikov section of X . Combining the model of Quillen and the model of Sullivan we obtain
MAHMOUD BENKHALIFA
Theorem 3.4.
Let X be a simply connected rational elliptic space of formal di-mension n such that n > m = max (cid:8) k | π k ( X ) = 0 (cid:9) and E ( X ) is finite. Then E ( X ) ∼ = . . . ∼ = E ( X m +2 ) ∼ = E ( X m +1 ) (15) Proof.
By hypothesis the Quillen model of X has the form ( L ( W ≤ n − ) , δ ), where W n − = 0 and its is Sullivan model has the form (Λ V ≤ m , ∂ ), where V m = 0. Recallthat V ∗ ∼ = Hom (cid:16) H ∗− ( L ( W ≤ n − )) , Q (cid:17) (16)Since n > m and by (16), we deduce that H k ( L ( W ≤ n − )) = 0 , k ≥ m. (17)it follows H n − ( L ( W ≤ n − )) = 0 . (18)Let us consider the Whitehead exact sequence of ( L ( W ≤ n − ) , δ ), namely · · · → H k ( L ( W ≤ n − )) → W k b k −→ H k − ( L ( W ≤ k − )) → H k − ( L ( W ≤ n − )) → · · · (19)which implies · · · → W n = 0 b n −→ H n − ( L ( W ≤ n − )) → H n − ( L ( W ≤ n − )) = 0 → · · · As a result, we obtain H n − ( L ( W ≤ n − )) = 0 (20)Now, according to (17), the map b n − is an isomorphism and according to Corollary2.3, it follows that R n − n − ∼ = E ( L ( W ≤ n − )) . Using Theorem 2.2 (for q = n − k = n −
2) and (20), we conclude that E ( L ( W ≤ n − )) ∼ = R n − n − . Consequently E ( L ( W ≤ n − )) ∼ = E ( L ( W ≤ n − ))Using the same arguments and taking in account the relation (17) which impliesthat b k is an isomorphism for k ≥ m + 1, by iterating the above process it followsthat E ( L ( W ≤ n − )) ∼ = E ( L ( W ≤ n − )) ∼ = . . . ∼ = E ( L ( W ≤ m ))Finally, by the properties of the Sullivan and Quillen models and taking into con-sideration Theorem 3.3 we obtain (15) (cid:3) Theorem 3.5.
Let X be a -connected rational elliptic space of formal dimension n . If π n ( X ) = 0 , then the group E ( X ) is infinite.Proof. Since the formal dimension of X is n , we can choose ( L ( W ≤ n − ) , δ ) as theQuillen model of X with W n − = 0 and (Λ V ≤ n − , ∂ ) as its Sullivan model.Next, taking q = n − k = n − (cid:0) W n − , H n − ( L ( W ≤ n − )) (cid:1) E ( L ( W ≤ n − )) ։ R n − n − (21)Assume by contradiction that E ( X ) is finite. By Theorem 2.12, we have E (Λ V ≤ n − ) ∼ = E (Λ V ≤ n ), so E (Λ V ≤ n ) is also finite and the short exact sequence (14) implies that H n (Λ V ≤ n − ) = 0 because V n ∼ = π n ( X ) = 0. Taking in account that X is 2-connected, the formula (11) implies that H n (Λ V ≤ n − ) ∼ = H n − ( L ( W ≤ n − )). So H n − ( L ( W ≤ n − )) = 0. Now recall that R n − n − is the subgroup of aut( W n − ) × E ( L ( W ≤ n − )) consisting of the pairs ( ξ, [ α ]) making the following diagram com-mutes W n − ✲ W n − b n − b n − ❄ ❄ ξH n − ( α ) H n − ( L ( W ≤ n − )) ✲ H n − ( L ( W ≤ n − ))As H n − ( L ( W ≤ n − )) = 0, we deduce that R n − n − = aut( W n − ) × E ( L ( W ≤ n − ))implying that R n − n − is infinite and by (21) the group E ( L ( W ≤ n − )) is also infinitecontradicting the fact that E ( X ) ∼ = E ( L ( W ≤ n − )) is finite. (cid:3) Self-closeness number N E ( X ) . For a space X , let [ X, X ] denote the monoidof homotopy classes of self-maps of the space X and let A k ( X ) = n f ∈ [ X, X ] | π i ( f ) : π i ( X ) ∼ = −→ π i ( X ) for any i ≤ k } In [14], Choi and Lee introduced the following concept:
Definition 3.6.
The self-closeness number, denoted by N E ( X ), is defined as fol-lows N E ( X ) = min n k | A k♯ ( X ) = E ( X ) o Theorem 3.7. If X is a simply connected rational elliptic space of formal dimen-sion n , then N E ( X ) ≤ n Proof. As X is of formal dimension n , by remark 3.2 its Sullivan model has theform (Λ V, ∂ ) = (Λ V ≤ n − , ∂ ). Define A k (Λ V ) = n [ α ] ∈ [Λ V, Λ V ] | e α i : V i ∼ = −→ V i for any i ≤ k o , where [Λ V, Λ V ] denotes the monoid of homotopy cochain algebras of self-equivalencesclasses of Λ V and where e α is the graded linear isomorphism induced by α on thegraded vector space of indecomposables V (see ([18] 12,15.(d)).Clearly by virtue on the properties of the Sullivan model we can identify the twosets A k (Λ V ) and A k ( X ).First we have A n − (Λ V ) = E (Λ V ) and A n (Λ V ≤ n ) = E (Λ V ≤ n ). Next it is easyto see that the set A n (Λ V ≤ n ) can be identified as a subset of A n − (Λ V ) byconsidering the following injective map θ : A n (Λ V ≤ n ) ֒ → A n − (Λ V ) , [ α ] θ ([ α ]) = [ β ] (22)where β = α on V ≤ n and β = id on V >n . Finally using Theorem 3.3 and (22) weget A n (Λ V ≤ n ) ⊆ A n − (Λ V ) = E (Λ V ) = E (Λ V ≤ n ) = A n (Λ V ≤ n )Therefore A n (Λ V ≤ n ) = E (Λ V ) implying that N E (Λ V ) ≤ n . (cid:3) Example 3.8.
Given a finite group G . According to a Theorem of Frucht [19],there exists a connected finite graph G = ( V, E ), where V denotes the set of the vertices of G and E the set of its edges, such that aut( G ) ∼ = G . Recall that in [15]Costoya and Viruel constructed a free commutative differential graded algebra (cid:0) Λ( x , x , y , y , y , z, { z v , x v } v ∈ V ) , ∂ (cid:1) where the degrees of the elements are | x | = 8 , | x | = 10 , | x v | = 40 , | z | = | z v | = 119 | y | = 33 , | y | = 35 , | y | = 37and where the differential is given by ∂ ( x ) = ∂ ( x ) = ∂ ( x v ) = 0 , ∂ ( y ) = x x , ∂ ( y ) = x x , ∂ ( y ) = x x ∂ ( z v ) = x v + X ( v,w ) ∈ E x v x w x ,∂ ( z ) = y y x x − y y x x + y y x + x + x and proved that E (Λ( x , x , y , y , y , { z v , w v } v ∈ V )) ∼ = G . Moreover they showedthat (cid:0) Λ( x , x , y , y , y , { z v , w v } v ∈ V ) , ∂ (cid:1) is an elliptic of formal dimension n =208 + 80 | V | with | V | the order of the graph G .Let X be a rational space whose admits (cid:0) Λ( x , x , y , y , y , z, { z v , x v } v ∈ V ) , ∂ (cid:1) as Sullivan model. On one hand and as X has formal dimension n = 208+80 | V | theQuillen model of X can be written as ( L ( W ≤ | V | ) , δ ) and because X = X [120] we deduce that N E ( X ) ≤ m = max (cid:8) k | π k ( X ) = 0 (cid:9) = max (cid:8) k | V k = 0 (cid:9) = 119 . Therefore applying Theorem 3.4 leads to E ( X ) ∼ = E ( X | V | ) ∼ = . . . ∼ = E ( X ) ∼ = G Therefore we can ameliorate Costoya and Viruel Theorem by reducing the formaldimension of X showing that every finite group G occurs as G = E ( X ) for somerational space X having formal dimension n = 120. Remark . It is important to notice that the space Z = X is not ellip-tic. Indeed, if Z were elliptic, H ∗ ( Z, Q ) would be Poincar´e duality ([18], Theo-rem A), and thus dim ( H ( Z, Q )) = 1. But an easy computation shows thatdim ( H ( Z, Q )) ≥ | V | > Example 3.10.
In ([1], Example 5.2), Arkowitz and Lupton constructed a freecommutative differential graded algebra (cid:0) Λ( x , x , y , y , y , z ) , ∂ (cid:1) with | x | = 10, | x | = 12, | y | = 41, | y | = 43, | y | = 45 and | z | = 119. The differential is given by ∂ ( x ) = ∂ ( x ) = 0 , ∂ ( y ) = x x , ∂ ( y ) = x x , ∂ ( y ) = x x ∂ ( z ) = y y x − y y x x + y y x x + x + x and proved that E (Λ( x , x , y , y , y , z ) ∼ = Z .Moreover they showed that (cid:0) Λ( x , x , y , y , y , z ) , ∂ (cid:1) is an elliptic of formal dimen-sion 188.If X is a rational space whose admits (cid:0) Λ( x , x , y , y , y , z ) , ∂ (cid:1) as Sullivan model,then its Quillen model can be written as ( L ( W ≤ ) , δ ). Since m = max (cid:8) k | π k ( X ) =0 (cid:9) = max (cid:8) k | V k = 0 (cid:9) = 119 . Therefore applying Theorem 3.4 leads to E ( X ) ∼ = E ( X ) ∼ = . . . ∼ = E ( X ) ∼ = Z Example 3.11.
Define (Λ( x, y ) , ∂ ) with | x | = 2 p . The differential is given by ∂ ( x ) = 0 , ∂ ( y ) = x a , a ≥ x, y ) , ∂ ) is elliptic and it easy to see that its formal dimension is n = 2( a − p and m = max (cid:8) k | V k = 0 (cid:9) = 2 ap − , so m > n . Now if X is arational space having (Λ( x, y ) , ∂ ) as the Sullivan model, then by applying Theorem3.3 we get E ( X ) ∼ = E (Λ( x )) ∼ = Q − { } . References
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