On strongly inflexible manifolds
aa r X i v : . [ m a t h . G T ] J a n ON STRONGLY INFLEXIBLE MANIFOLDS
CRISTINA COSTOYA, VICENTE MU ˜NOZ, AND ANTONIO VIRUEL
Abstract.
An oriented compact connected N -manifold M is inflexible if it does notadmit self-maps of unbounded degree. In addition, if all the maps from any other orientedcompact connected N -manifold have bounded degree, then M is said to be stronglyinflexible. The existence of inflexible manifolds was established by Arkowitz and Lupton,however no example of simply-connected strongly inflexible manifold is known. We provethat all but one of the known examples of simply-connected inflexible manifolds are non-strongly inflexible. Introduction
Let M be an oriented compact connected N -manifold. We are interested in the setdeg( M ) = { deg( f ) | f : M → M continuous } Note that always 0 , ∈ deg( M ), and − ∈ deg( M ) if and only if M admits orientation-reversing self-maps. Note also that deg( M ) is a multiplicative semi-group, so if deg( M )contains any ℓ ∈ deg( M ) with | ℓ | >
1, then deg( M ) is unbounded. We say that M is inflexible if deg( M ) is bounded. It is known that hyperbolic manifolds are inflexible (see[9, Remark 6.2]. Moreover, for simply-connected examples, we have the work of [1, 8].Suppose that M , M are oriented N -manifolds, and letdeg( M, M ) = { deg( f ) | f : M → M continuous } . This set contains 0 ∈ deg( M, M ). If it contains a non-trivial element d ∈ deg( M, M ) and M is not inflexible, then the set deg( M, M ) is unbounded. We say that M is stronglyinflexible if for every N -manifold M the set deg( M, M ) is bounded. The bound dependson M , for instance, taking M = n M , there is a degree n map f : M → M . Observe thatif M is strongly inflexible, then M is automatically inflexible (unless deg( M, M ) = { } ).When M is strongly inflexible, a semi-norm can be defined | M | := max (cid:8) deg( M, M ) (cid:9) satisfying that, for given a map g : M → M ′ , | deg( g ) | | M ′ | ≤ | M | .The condition of being inflexible or strongly inflexible can be translated to the lan-guage of Sullivan algebras for simply-connected (or even nilpotent) spaces. Let (Λ V, d )be a Sullivan model corresponding to a Poincar´e duality space with formal dimension N . Mathematics Subject Classification.
Primary 55P62. Secondary 57N65, 55P10.The first author was partially supported by MINECO (Spain) grant MTM2016-79661-P. The secondauthor was partially supported MINECO (Spain) grant PGC2018-095448-B-I00. The third author waspartially supported by MINECO (Spain) grant MTM2016-78647-P.
Suppose that V = 0, and let ν ∈ (Λ V ) N be a representative of the volume form. Then(Λ V, d ) is inflexible if for every dga map ϕ : (Λ V, d ) → (Λ V, d )we have deg( ϕ ) = 0 , ±
1, where H ([ ν ]) = deg( ϕ )[ ν ]. Moreover, the Sullivan algebra (Λ V, d )is strongly inflexible if for any Sullivan model (Λ
W, d ), corresponding to a Poincar´e dualityspace with formal dimension N , the set of degrees of all the dga morphisms(Λ V, d ) → (Λ W, d )is finite.As we have mentioned in the abstract, no example of strongly inflexible manifold (ordga) is known to date. In this paper we give a sufficient condition, Corollary 2.12, for beingnon-strongly inflexible: There exists a dga (Λ
W, d ) with a positive weight (Definition 2.1)and a morphism ϕ : (Λ V, d ) → (Λ W, d )of non-zero degree, that is H ( ϕ )([ ν ]) = 0.Using the above condition, we prove that all , but one, of the known examples of in-flexible dgas and manifolds, as given in [1, 2, 6, 7, 8, 9], are non-strongly inflexible (seeTheorem 5.2 and Theorem 5.4). For the one example for which we have not yet been ableto prove that is non-strongly inflexible see Remark 5.5.We raise the following conjecture: Conjecture 1.1.
There do not exist strongly inflexible simply-connected manifolds. Weights
In this section we discuss the key ingredient in this work, which is the relationshipbetween weights in a dga and the strongly inflexibility property.Our definition of weight is taken from [3, 4].
Definition 2.1.
Let A := ( A , d ) be a dga. We say that A has a weight ω if each degreecomponent A k of A decomposes as a direct sum A k = M n ∈ Z n A k , k ≥ , such that (i) d ( n A k ) ⊂ n A k +1 , and (ii) n A k ∧ m A l ⊂ n + m A k + l .Every non zero element x ∈ n A = L k ≥ n A k is said to have weight n , and it is denoted by ω ( x ) = n . An element x ∈ A is said ω -homogeneous if x is a sum of monomials havingthe same weight. Some useful notions of weights.
Definition 2.2.
Let A be a a dga with a weight ω . (i) Given a ∈ A we say that ω detects a if a is ω -homogeneous and ω ( a ) = 0 . N STRONGLY INFLEXIBLE MANIFOLDS 3 (ii)
We say that ω is non trivial if it detects some non zero element a ∈ A .Remark . Let A be a a dga with a weight ω .(i) The constants A = Q have always weight ω ( λ ) = 0, for λ ∈ A .(ii) The weight ω is positive if n A = 0 for every n < A = A .(iii) The weight ω on A induces a weight, also denoted ω , in the cohomology of A . Remark . Let A be a dga with a weight ω . If ω is positive, then ω detects every ω -homogeneous element in A . Lemma 2.5.
Let ( A , d ) be a dga with a weight ω . Given x ∈ H ( A , d ) , there exists adecomposition x = P [ a i ] , where the elements a i ∈ A are ω -homogeneous, closed and ω ( a i ) = ω ( a j ) if and only if i = j .Proof. Let x = [ a ] ∈ H ∗ ( A , d ) be a cohomology class. Decompose a = P a i into ω -homogeneous elements such that ω ( a i ) = ω ( a j ) if and only if i = j . Then 0 = da = P da i .As da i are ω -homogeneous, we have da i = 0. Hence [ a ] = P [ a i ]. (cid:3) Corollary 2.6.
Let ( A , d ) be a dga with a weight ω . If ( A , d ) has Poincar´e dualitycohomology, formal dimension N and volume form [ ν ] , then the representative ν can bechosen to be ω -homogeneous.Proof. According to Lemma 2.5, we have [ ν ] = P [ ν i ], where the elements ν i ∈ A are ω -homogeneous, closed and ω ( ν i ) = ω ( ν j ) if and only if i = j . Since H N ( A , d ) ∼ = Q , and H N ( A , d ) has a weight, it must be that only one i has [ ν i ] = 0. Hence [ ν ] = [ ν i ] and ν i is ω -homogeneous. (cid:3) Definition 2.7.
Let ( A , d ) be a dga. We say that a class x ∈ H ( A , d ) is flexible if forevery n ∈ Z there exists a dga morphism f n : A → A such that H ( f n )( x ) = λ x with λ ≥ n . The existence of non trivial weights implies the existence of flexible classes.
Lemma 2.8.
Let ( A , d ) be a dga, and x = [ a ] ∈ H ( A , d ) . If ( A , d ) has a weight ω thatdetects a , then x = [ a ] is a flexible class.Proof. For q ∈ Q , let ϕ q : ( A , d ) → ( A , d ) be the following morphism. If y an ω -homogeneous element, ϕ q ( y ) = q ω ( y ) y. Otherwise, we decompose y = P y i into ω -homogeneous elements and ϕ q ( y ) = P q ω ( y i ) y i . Now, given n ∈ Z , we construct f n : A → A such that H ( f n )( x ) = λ x with λ ≥ n asfollows. Since ω detects a , then ω ( a ) = 0 and therefore there exists q n ∈ Q such that q ω ( a ) n ≥ n . We define f n = ϕ q n . (cid:3) Corollary 2.9.
Let (Λ V, d ) be the Sullivan model of ( A , d ) , a dga with Poincar´e dualitycohomology, formal dimension N and volume form [ ν ] . If ( A , d ) has a weight ω thatdetects ν , then (Λ V, d ) is not inflexible.Proof. Since ω detects ν , then by Lemma 2.8 the volume form [ ν ] is a flexible class.Therefore, for every n ∈ Z there exists a dga morphism f n : A → A such that deg( f n ) = C. COSTOYA, V. MU ˜NOZ, AND A. VIRUEL λ ≥ n . Finally, according to the Lifting Lemma [10, Lemma 12.4], f n lifts to e f n , a self-map of (Λ V, d ) whose degree is deg( e f n ) = deg( f n ) = λ ≥ n , and therefore (Λ V, d ) is notinflexible. (cid:3)
Proposition 2.10.
Let ( A , d ) be a simply-connected, finite type dga. For any non-zeroclass [ ν ] ∈ H N ( A , d ) , there is a dga ( ¯ A , d ) whose cohomology is Poincar´e duality of formaldimension N , and a dga map q : ( A , d ) → ( ¯ A , d ) , such that H ( q )([ ν ]) = 0 is a volume form for H ∗ ( ¯ A , d ) .Proof. We take a simply-connected finite type CW-complex X whose Sullivan model isisomorphic to ( A , d ), [16, Theorem 10.2 (ii)]. Let X ′ be the ( N + 1)-skeleton of X .Therefore H ≤ N ( X ′ ) ∼ = H ≤ N ( X ) ∼ = H ≤ N ( A , d ). This is a finite CW-complex. By atheorem of Thom [15, Th´eor`emes III.4], the cohomology class [ ν ] ∈ H N ( X ′ ) is detectedby an oriented compact N -manifold Z , that is there is a map f : Z → X ′ such that h [ ν ] , f ∗ [ Z ] i = h f ∗ [ ν ] , [ Z ] i .We need to arrange that Z is simply-connected, and can do this by a surgery process.Let us assume for the moment that N ≥
3. As Z is compact, the fundamental groupis finitely generated. Let γ i be loops generating π ( Z ), and we can perturb them sothat they are embedded and disjoint. Take a small tubular neighbourhood of γ i , which is U i ∼ = S × D N − , since Z is oriented. Take a smaller disc D ′ N − ⊂ D N − and homotop f sothat it is constant in the normal directions on U ′ i = S × D ′ N − , that is f ( x, y ) = γ i ( x ), for( x, y ) ∈ U ′ i = S × D ′ N − . The loop f ◦ γ i : S → X ′ is contractible, and let H i : D i → X ′ be a contraction. Now we surger out the loops γ i , that is we consider the N -manifold Z ′ := ( Z − ∪ U ′ i ) ∪ ( ∪ D i × ∂D ′ N − ) . A Seifert-van Kampen argument shows that Z ′ is simply-connected, and f can be extendedto a map f ′ : Z ′ → X ′ , with f ′∗ [ Z ′ ] = f ∗ [ Z ].Let ( ¯ A , d ) be a model of Z ′ , that exists since Z ′ is simply-connected. The cohomology of( ¯ A , d ) is Poincar´e duality of formal dimension N . Then f ′ : Z ′ → X ′ ⊂ X is representedby a dga map q : ( A , d ) → ( ¯ A , d ) such that H ( q )([ ν ]) = 0. The case N = 2 can be doneby hand, since we can take ( ¯ A , d ) to be the model of S . (cid:3) Remark . The assumption of simply-connectedness in Proposition 2.10 is not nec-essary. The space X can be taken a nilpotent space. Then we can surger Z so that π ( f ) : π ( Z ′ ) → π ( X ′ ) is injective, by killing the generators of the kernel of the mor-phism on fundamental groups. Thus Z ′ is nilpotent, and hence it admits a model. Corollary 2.12.
Let ( A , d ) be a dga with Poincar´e duality and volume form [ ν ] ∈ H N ( A , d ) . Suppose that there exists a dga ( A ′ , d ) and a map ϕ : ( A , d ) → ( A ′ , d ) suchthat H ( ϕ )([ ν ]) = 0 . If ( A ′ , d ) has a positive weight ω then ( A , d ) is non-strongly inflexible.Proof. By Lemma 2.5, there exists a decomposition H ( ϕ )([ ν ]) = r P i =0 [ a ′ i ] such that a ′ i are ω -homogeneous, closed and w ( a ′ i ) = w ( a ′ j ) if i = j . Assume that ω ( a ′ ) = max { ω ( a ′ i ) | i =0 , . . . , r } .By Lemma 2.8, for every n , there is a map f n : ( A ′ , d ) → ( A ′ , d ) verifying that H ( f n )[ a ′ i ] = q ω ( a ′ i ) n [ a ′ i ] for every i , where q n is a natural number satisfying that q ω ( a ′ ) n ≥ n . N STRONGLY INFLEXIBLE MANIFOLDS 5
Now, associated to ( A ′ , d ), by Proposition 2.10 there is a dga ( ¯ A , d ) whose cohomologyis Poincar´e duality of formal dimension N , and a dga map q : ( A ′ , d ) → ( ¯ A , d ) such that H ( q )([ a ′ ]) = 0 is a volume form, let us call it [ µ ], for H ∗ ( ¯ A , d ).Finally, take G n = q ◦ f n ◦ ϕ : ( A , d ) → ( ¯ A , d ), which in cohomology: H ( G n )[ ν ] = ( q ω ( a ′ ) n + r X i =1 q ω ( a ′ i ) n α i )[ µ ] , α i ∈ Q . Using that q ω ( a ′ ) n ≥ n and that ω ( a ′ ) = max { ω ( a ′ i ) | i = 1 , . . . , r } , we deduce that the set { deg( G n ); n ∈ Z } is unbounded and therefore ( A , d ) is non-strongly inflexible. (cid:3) Proposition 2.13.
Let (Λ V, d ) be a -step Sullivan algebra. Then it admits a positiveweight.Proof. A 2-step Sullivan algebra is of the form Λ V for V = V ⊕ V such that d ( V ) = 0, and d ( V ) ⊂ Λ V . Let { x i } be a homogeneous basis of V and { y j } a homogeneous basis of V .Then a positive weight ω is defined by declaring the elements of the basis ω -homogeneousof weight ω ( x i ) = | x i | , ω ( y j ) = | y j | + 1 . and extend this weight to monomials in Λ V according to the rule in Definition 2.1(ii).It remains to check that Definition 2.1(i) also holds. Indeed, given a monomial z = (cid:0) Y i ∈ I x n i i (cid:1)(cid:0) Y j ∈ J y m j j (cid:1) , it is ω -homogeneous of weight ω ( z ) = X i ∈ I n i | x i | + X j ∈ J m j ( | y j | + 1) , while dz = X k ∈ J ǫ k m k ( dy k ) y m k − k (cid:0) Y i ∈ I x n i i (cid:1)(cid:0) Y j ∈ J −{ k } y m j j (cid:1) , where ǫ k = ± dy k ∈ V . Therefore ω ( dz ) = ω (cid:16) ( dy k ) y m k − k (cid:0) Y i ∈ I x n i i (cid:1)(cid:0) Y j ∈ J −{ k } y m j j (cid:1)(cid:17) = ( | y k | + 1) + ( m k − | y k | + 1) + X i ∈ I n i | x i | + X j ∈ J −{ k } m j ( | y j | + 1)= ω ( z ) , and the condition in Definition 2.1(i) holds. (cid:3) Corollary 2.14.
Every -step Sullivan algebra is non-strongly inflexible. C. COSTOYA, V. MU ˜NOZ, AND A. VIRUEL Universal spaces
In this section we develop the necessary tools to extend our previous results on non-strongly inflexibility from dgas to manifolds.Recall that given p , a prime or zero, a map f : X → Y is said to be a p -equivalence if f induces an isomorphism on H ∗ ( X ; Z /p ) ∼ = H ∗ ( Y ; Z /p ). Here Z / Q .Combining [5, 13, 14], a finite CW-complex X is said to be universal if for any p , forany given p -equivalence k : Y → Z , and for an arbitrary map g : X → Z , there is a map h : X → Y and there is a p -equivalence f : X → X such that the following diagramcommutes up to homotopy: Y k / / ZX h O O f / / X g O O Universal spaces are characterized by their minimal models ([5, Theorem A]):
Theorem 3.1.
Let X be a simply-connected finite CW-complex, and (Λ V, d ) be its mini-mal model. Then X is universal if and only if (Λ V, d ) admits a positive weight. The cohomology of universal spaces is generated by flexible classes.
Theorem 3.2.
Let M be a manifold with volume form η ∈ H N ( M ; Z ) . Assume that thereexits a map g : X → M such that X is a simply-connected finite universal CW-complex, H N ( X ; Q ) ∼ = Q , and g ∗ ( η ) ⊗ Q = 0 . Then M is non-strongly inflexible.Proof. Let (Λ
V, d ) be the Sullivan minimal model of X , which by Theorem 3.1 admits apositive weight ω. Then, by Lemma 2.5, there exists a decomposition of g ∗ ( η ) ⊗ Q ω -homogeneous elements: g ∗ ( η ) ⊗ Q v ] ∈ H N ( X ; Q ) ∼ = H N (Λ V, d ) ∼ = Q . Using Lemma 2.8, for any n ∈ N , we choose an integer l n such that l ω ( v ) n ≥ n . Then thereexists a dga morphism: f n : (Λ V, d ) → (Λ V, d ) v l ω ( v ) n v, for every ω -homogeneous element of (Λ V, d ).Let ζ X : X → X be the rationalization of X , that is, the localization at 0, and let ψ n : X → X be the realization of f n . Using that X is 0-universal, there exists a 0-equivalence e ψ n : X → X and a map h n : X → X such that the following diagramhomotopy commutes: X ζ X / / X ψ n / / X X e ψ n / / h n O O X ζ X O O (1)Observe that f ψ n , ψ n and ζ X are 0-equivalences, thus h n also is a 0-equivalence. N STRONGLY INFLEXIBLE MANIFOLDS 7
Using that (1) is homotopy commutative, we get that H N ( f ψ n ; Q )([ v ]) = H N ( ψ n ◦ ζ X ◦ h n ; Q )([ v ]) = H N ( h n ; Q )( l ω ( v ) n [ v ]) = c n l w ( v ) n [ v ] , (2)where c n is a non-zero integer since h n is a 0-equivalence.Let ν = g ∗ ( η ) ∈ H N ( X ; Z ). By [15, Th´eor`emes III.4], there exists an N -manifold M ′ with volume form µ , and a map θ : M ′ → X such that θ ∗ ( ν ) = kµ for some integer k = 0.We are going to show that deg( M ′ , M ) is unbounded, therefore M can not be stronglyinflexible.Indeed, for every n ∈ N , we can define the composition: G n : M ′ θ / / X e ψ n / / X g / / M Then, G ∗ n ( η ) = ( e ψ n ◦ θ ) ∗ ( g ∗ ( η )) = θ ∗ ( e ψ ∗ n ( ν )) (now we use (2))= θ ∗ ( c n l ω ( v ) n ν + t ) (where t is some torsion element)= c n l ω ( v ) n θ ∗ ( ν ) ( θ ∗ ( t ) must be trivial since H N ( M ′ ; Z ) ∼ = Z )= kc n l ω ( v ) n µ. Therefore | deg( G n ) | = | kc n l ω ( v ) n | ≥ n , as k and c n are non-zero integers. (cid:3) Proposition 3.3.
Let ( A , d ) be a connected dga with positive weight ω . Then thereexists a minimal model ρ : (Λ V, d ) ∼ −→A such that (Λ V, d ) has a positive weight e ω with ω ( ρ ( v )) = e ω ( v ) , for every e ω -homogeneous element v ∈ Λ V .Proof. First assume that A is simply-connected. We construct V n inductively. SupposeΛ V The case where A is not simply-connected is similar, but now for every degree we haveto do an induction, and the space V n is constructed as the direct limit of the inductivesystem. The weight behaves well by direct limits. (cid:3) Lemma 3.4. Let ( A , d ) be a connected dga with positive weight ω , and I ⊂ A be adifferential closed ideal (that is, dI ⊂ I ) generated (as vector space) by ω -homogeneouselements. Then e A = A /I is a connected dga with positive weight e ω defined by e ω (¯ a ) = ω ( a ) ,for every ω -homogeneous element a ∈ A such that ¯ a = 0 .Proof. Since I ⊂ A is a differential closed ideal, then e A = A /I is a connected dga, and itonly remains to prove that the weight e ω is well defined.Assume e ω is not well defined. Therefore there exist ω -homogeneous elements a l ∈ A , l = 1 , 2, such that ω ( a ) = ω ( a ) and ¯ a = ¯ a = 0. Then a − a ∈ I , and a − a = P x i where every x i ∈ I is ω -homogeneous, and ω ( x i ) = ω ( x j ) if and only if i = j . Moreover, a l is ω -homogeneous, l = 1 , 2, and ω ( a ) = ω ( a ). Hence a l = x i ( l ) , l = 1 , 2, and ¯ a = ¯ a = 0.This is a contradiction as we assumed ¯ a = ¯ a = 0. (cid:3) Finally, we obtain an integral version of Corollary 2.12 Theorem 3.5. Let ( M, η ) be a simply-connected N -manifold with minimal model (Λ V, d ) .Write the rational volume form as η ⊗ Q ν ] . Assume there exist a dga morphism ψ : (Λ V, d ) → ( A , d ) such that ( A , d ) is a simply-connected finite type dga that has apositive weight and H N ( ψ )([ ν ]) = 0 . Then M is non-strongly inflexible.Proof. By Lemma 2.5, H N ( ψ )([ ν ]) = P [ a i ] where every a i ∈ A is closed ω -homogeneous,and ω ( a i ) = ω ( a j ) if and only if i = j . Fix e a ∈ A , a nontrivial a i in the decomposi-tion above, take a complement A N = h e a i ⊕ W , where W is spanned by ω -homogeneouselements, and define I = A ≥ N +1 ⊕ W. Then I ⊂ A is a closed differential ideal generated (as vector space) by ω -homogeneouselements. By Lemma 3.4, e A = A /I is a finite type connected dga with positive weight andformal dimension N . Therefore, by Proposition 3.3, e A admits a minimal model (Λ W, d )with positive weight, which is the rational homotopy type of a simply-connected finiteCW-complex X , with H N ( X ; Q ) ∼ = Q . Moreover, since (Λ W, d ) has a positive weight, X is universal by Theorem 3.1.Let us consider the composition e ψ : Λ V ψ / / A / / / / e A , and let e Ψ : X → M be itsgeometrical realization. Observe that H N ( e ψ )([ ν ]) = 0 and so H N ( e Ψ)([ ν ]) = 0 . Now, for ζ M : M → M (resp. ζ X : X → X ) the rationalization of M (resp. of X ),which are 0-equivalences, since X is universal, there exists a commutative diagram X g (cid:15) (cid:15) f / / X e Ψ ◦ ζ X (cid:15) (cid:15) M ζ M / / M N STRONGLY INFLEXIBLE MANIFOLDS 9 where f is a 0-equivalence. We are going to show that g ∗ ( η ) ⊗ Q = 0 and therefore, byTheorem 3.2, M is non-strongly inflexible: g ∗ ( η ) ⊗ Q H N ( g ; Q )([ ν ]) = H N ( e Ψ ◦ ζ X ◦ f ; Q )([ ν ]) = H N ( f ; Q )( H N ( e Ψ)([ ν ])) = 0 . (cid:3) 4. 3 -step Sullivan algebras Positive weights in Sullivan algebras relate to non inflexibility and non strongly in-flexibility properties. We are going to exploit this aspect of positive weights to give asystematic way to check that certain 3-step Sullivan algebras are non-strongly inflexible.Our procedure applies to all but one of the inflexible Sullivan algebras known up to present(see Example 5.3 and Remark 5.5). Definition 4.1. We say that a Sullivan algebra M = (Λ V ⊗ Λ V ⊗ Λ V , d ) , is a -step algebra if d ( V ) ⊂ Λ V and d ( V ) ⊂ Λ( V ⊕ V ) . We are only interested in 3-step Sullivan algebras of the form M = (Λ V ⊗ Λ V ⊗ Λ( z ) , d ) . Out of such a Sullivan algebra M , we are going to describe how to construct a universal B , dga admitting positive weights, and a morphism of dgas ψ : M → B . Therefore, wehave a sufficient condition, Corollary 2.12, to check that M is non-strongly inflexible.We first introduce some notions. Definition 4.2. A derivation differential graded algebra, ddga for short, is a triple ( A , d, θ ) such that ( A , d ) is a dga, and θ is a derivation of degree on A such that dθ = θd .Let ( A , d, θ ) and ( B , d, θ ′ ) be ddga. A ddga morphism f : A → B is a dga morphismsuch that f ◦ θ = θ ′ ◦ f . Definition 4.3. Let ( A , d ) be a dga. We call the free derivation differential graded algebragenerated by A to a ddga ( P ( A ) , d, θ ) equipped with a dga morphism i : A → P ( A ) whichuniversal for ddgas, that is, for any ddga ( B , d, θ ′ ) and any dga map f : A → B , there isa unique ddga morphism ¯ f : P ( A ) → B such that f = ¯ f ◦ i . Clearly, as it is defined by an universal property, ( P ( A ) , d, θ ) has to be unique up tounique isomorphism. For its existence, it can be constructed as follows. Let b P ( A ) = T ( A ⊕ θ A ⊕ θ A ⊕ . . . )be the tensor algebra on elements of A and “formal” derivatives θ k A ∼ = θ ( θ k − ) A , for k ≥ 1, and θ A = A . There is a natural derivation θ on b P ( A ). Consider I ⊂ b P ( A ) theideal generated by elements a ⊗ b − ab , and θ ( ab ) − θa ⊗ b − a ⊗ θb , for any pair of elements a, b ∈ A . This is a differential ideal, that is θ ( I ) ⊂ I . Finally, set P ( A ) = b P ( A ) /I, and d ( θa ) = θ ( da ), for all a ∈ A . Unfortunately, the algebra P ( A ) is not of finite type. To solve this, we define the dotalgebra. Definition 4.4. Let ( A , d, θ ) be a ddga. We say A is dot algebra if θ = 0 . There is a notion of free dot algeba generated by a dga: Definition 4.5. Let ( A , d ) be a dga. We call the free dot algebra generated by A orsimply dot algebra of A , to a dot algebra ( ˙ A , d, θ ) equipped with a dga morphism i : A → ˙ A which is universal for dot algebras, that is for any dot algebra ( B , d, θ ′ ) and any dga map f : A → B , there is a unique ddga morphism ¯ f : ˙ A → B such that f = ¯ f ◦ i . The definition is ˙ A = P ( A ) /P ≥ ( A ) , which is the quotient of P ( A ) by the ideal generated by θ ( A ) and θ ( A ) · θ ( A ). We denote˙ a = θ ( a ) . Now let us deal with the situation of 3-step Sullivan algebras M = (Λ V ⊗ Λ V ⊗ Λ( z ) , d ).Denote A := Λ V = Λ( V ⊕ V ) = Λ( x i , y j ), where we denote the generators of V as x i , anddenote the generators of V as y j . We put the positive weight ω on A given by Proposition2.13. Let dz = P + P + . . . + P m be the decomposition into ω -homogeneous elements. If m = 0, then dz is ω -homogeneousand we can define ω ( z ) = ω ( P ), and M has a positive weight. Hence it is non-stronglyinflexible.Now we shall assume that dz = P + P . That is, we focus on the case m = 1, which isall that we need for our applications. Note that all P , P , . . . , P m are all of them closedelements. Then, the dot algebra of A = Λ( V ⊕ V ) is actually˙ A = A ⊗ ( Q ⊕ ˙ V ⊕ ˙ V ) . Let us introduce elements u , u , u with differentials given by: du = P + ˙ P ,du = P ,du = ˙ P . We assign weight to the dot algebra given by ω ( ˙ y ) = ω ( y ) + ( ω ( P ) − ω ( P )) , so that P + ˙ P is ω -homogeneous. Now we set the weights ω ( u ) = ω ( P + ˙ P ) , ω ( u ) = ω ( P ) , ω ( u ) = ω ( ˙ P ) . This provides a weight to B := ˙ A ⊗ Λ( u , u , u ). Finally consider the dga map ψ : M = A ⊗ Λ( z ) −→ B = ˙ A ⊗ Λ( u , u , u ) ,a ∈ A 7→ a + ˙ a,z u + u + u . (3)With the same notation as above, we have the following: N STRONGLY INFLEXIBLE MANIFOLDS 11 Proposition 4.6. Let [ ν ] ∈ H N ( A ) be a non-zero cohomology class and let n = N − | dz | .Suppose that there is not any non-zero α ∈ H n ( A ) such that α ∪ [ P ] = 0 , α ∪ [ P ] = 0 .If H ( ψ )([ ν ]) = 0 then there exist closed elements A, B ∈ A d such that we can write [ ν ] = [ AP + BP ] in H ( A ) and ( A − B ) ˙ P ∈ (cid:0) ( P , ˙ P , P ) ∩ ker d (cid:1) + im d, as elements in the algebra ˙ A .Proof. If H ( ψ )([ ν ]) = 0 then ν + ˙ ν = dχ is exact in B . Write χ = X ( A i + ˙ B i ) u i + ( C ij + ˙ D ij ) u i u j + ( E + ˙ F ) u u u + ( G + ˙ H ) , for some elements A i , ˙ B i , C ij , ˙ D ij , E, ˙ F , G, ˙ H in ˙ A , where the dot means that the elementlies in the “dot” part, and the indices i < j .Then we have the following set of equations ν = A P + A P + dG, ˙ ν = ˙ B P + A ˙ P + ˙ B P + A ˙ P + d ˙ H, dA − C P , dA + C P , dC , (4)the last three obtained by taking the coefficients of u , u , u u , respectively, in the equality ν + ˙ ν = dχ .From the above, α = [ C ] is a cohomology class with α ∪ [ P ] = 0, α ∪ [ P ] = 0. Byhypothesis, C = dη for some η . Hence A = A − ηP , B = A + ηP are closed and ν = AP + BP + dG .Now apply the dot operator to the first equation in (4), to get˙ ν = ˙ A P + ˙ A P + A ˙ P + A ˙ P + d ˙ G, and compare it with the second equation, to get that ( A − A ) ˙ P ∈ ( P , ˙ P , P ) + im d .Clearly ( A − B ) ˙ P also lies in the same ideal, and moreover, it is closed, so it belongs alsoto ker d . (cid:3) Under the same assumptions and notation, we can now prove the main result in thissection: Theorem 4.7. Let [ ν ] ∈ H N ( A ) be a non-zero cohomology class and let n = N − | dz | .Suppose that there is not any non-zero α ∈ H n ( A ) such that α ∪ [ P ] = 0 , α ∪ [ P ] = 0 ,and that whenever we have closed elements A, B ∈ A n such that [ ν ] = [ AP + BP ] , wealso have ( A − B ) ˙ P / ∈ (cid:0) ( P , ˙ P , P ) ∩ ker d (cid:1) + im d. Then M = A ⊗ Λ( z ) is non-strongly inflexible.Let ( M, η ) be a simply-connected N -manifold for which M is a model. Then M isnon-strongly inflexible. Proof. By Proposition 4.6, we deduce that H ( ψ )([ ν ]) = 0, where ψ is the map (3). As B admits a positive weight, then ω detects H ( ψ )([ ν ]). We apply Corollary 2.12 to concludethat M is non-strongly inflexible. The last statement follows from Theorem 3.5. (cid:3) The Arkowitz-Lupton Example The existence of inflexible manifolds was first established by Arkowitz and Lupton in [2,Example 5.1, Example 5.2]. They gave examples of simply-connected Sullivan algebrasthat have finitely many homotopy classes of dga endomorphisms. Then, using Bargeand Sullivan obstruction theory, they showed that those examples are minimal models ofsimply-connected manifolds. In particular, these manifolds are inflexible.The succeeding examples in literature have been built upon [2, Example 5.1] that weintroduce now: Definition 5.1. (The Arkowitz-Lupton example) Let M = (Λ( x , x , y , y , y , z ) , d ) bethe Sullivan algebra with | x | = 8 dx = 0 | x | = 10 dx = 0 | y | = 33 dy = x x | y | = 35 dy = x x | y | = 37 dy = x x | z | = 119 dz = x γ + x + x where γ = αβ , α = x y − x y , β = x y − x y . So γ = x y y − x x y y + x y y . In the following we are going to show that the dga from Definition 5.1 is non-stronglyinflexible. Observe that M is a 3-step algebra as we introduced in the previous section.With the same notation as in Section 4, the 2-step algebra A = (Λ( x , x , y , y , y ) , d ) , has a positive weight ω given by Proposition 2.13. The cohomology is H ∗ ( A ) = h x n , x m , x n α, x m α, x n β, x m β, x n γ, x m γ | n, m ≥ i⊕ h x x , x x , x x , x x α, x x β, x x α i . Note that there is a “free” part (first summand) and some extra low degree terms gener-ators. There are the following exact elements: d ( y y ) = x x α, d ( y y ) = x x β, d ( y y ) = x x β + x x α. (5)We also introduce the dot algebra, Definition 4.5,( ˙ A , d ) = A ⊗ ( Q ⊕ ˙ V ⊕ ˙ V ) , where N STRONGLY INFLEXIBLE MANIFOLDS 13 | ˙ x | = 8 d ˙ x = 0 | ˙ x | = 10 d ˙ x = 0 | ˙ y | = 33 d ˙ y = 3 x ˙ x x + x ˙ x | ˙ y | = 35 d ˙ y = 2 x ˙ x x + 2 x x ˙ x | ˙ y | = 37 d ˙ y = ˙ x x + 3 x x ˙ x and ˙ γ = ˙ αβ + α ˙ β , with ˙ α = ˙ x y + x ˙ y − ˙ x y − x ˙ y , ˙ β = ˙ x y + x ˙ y − ˙ x y − x ˙ y .Therefore ˙ γ = 2 ˙ x x y y − ˙ x x y y − x ˙ x y y + 2 x ˙ x y y + x ˙ y y − x x ˙ y y + x ˙ y y + x y ˙ y − x x y ˙ y + x y ˙ y (6)We have assembled the necessary elements to apply Theorem 4.7 and prove our mainresult in this section. For the proof to be more readable, we have moved some technicallemmas that are needed to the end of this section. Theorem 5.2. The dga M from Definition 5.1 is non-strongly inflexible. Furthermore,any manifold for which M is a Sullivan model is also non-strongly inflexible.Proof. Let us take [ ν ] = [ x ] ∈ H ( A ), N = 208. Now dz = P + P , where P = x + x ,P = x γ. Now, n = N − | dz | = 208 − 120 = 88 and the cohomology H ( A ) = h [ x ] , [ γ ] i . For any α ∈ H ( A ) satisfying α ∪ [ P ] = 0 and α ∪ [ P ] = 0 it is clear that α = 0.Let us write [ ν ] = [ AP + BP ], where A, B ∈ A are closed elements. The possiblesolutions are of the form A = x + dη A B = aγ + dη B for a ∈ Q . Then, we deduce that( A − B ) ˙ P = (cid:0) x ˙ x γ + x ˙ γ (cid:1) + d (( η A − η B ) ˙ P )using that ˙ P = 4 x ˙ x + x ˙ γ , γ = 0 and γ ˙ γ = 0.In order to apply Theorem 4.7, we need to check that the element ( A − B ) ˙ P is not inthe ideal I = (cid:0) ( x + x , x ˙ x + 12 x ˙ x , x γ ) ∩ ker d (cid:1) + im d ⊂ ˙ A . Since 4 x ˙ x γ ∈ I, it is enough to check that X := x ˙ γ / ∈ I .First, notice that there is a y -gradation in ˙ A according to the number of y j , ˙ y j , andthe differential lowers the degree by one. The y -degree of X is two, so assume for acontradiction that X = Y + dη , for some closed element Y in I of y -degree equals two.Namely Y = ˙ F ( x + x ) + G (15 x ˙ x + 12 x ˙ x ) + ˙ Hx γ, (7) where ˙ F , G, ˙ H are of the form˙ F = ˙ F ′ + ˙ F ′′ , ˙ F ′ = X F ′ ij y i ˙ y j , ˙ F ′′ = X F ′′ ijk ˙ x k y i y j ,G = X G ij y i y j , ˙ H = X H k ˙ x k , with F ′ ij , F ′′ ijk , G ij , H k ∈ Q [ x , x ] . As Y is closed, we have0 = dY = d ( ˙ F ′ + ˙ F ′′ )( x + x ) + dG (15 x ˙ x + 12 x ˙ x ) . Look at the ˙ y j -term in this expression, j = 1 , 2, to conclude that P i F ′ ij y i is closed. Hence,according to degrees, we deduce that˙ F ′ ∈ W = h x β ˙ y , x α ˙ y , x β ˙ y , x α ˙ y i . Recalling that X = Y + dη , and (5), we see that the ˙ y -part of the exact term dη is ofthe form x x P ( R j α + S j β ) ˙ y j , with R j , S j elements in Q [ x , x ] . Looking at the components x α, x α, x β, x β , and comparing the ˙ y j -parts, j = 1 , , X , Y and dη , we obtain that: ˙ F ′ = x β ˙ y + x α ˙ y . Therefore using (6) and (7), the equality X = Y + dη becomes x ˙ γ = x (cid:0) − βx ˙ y + ( βx − αx ) ˙ y + αx ˙ y (cid:1) + x (cid:0) ( βy + αy ) ˙ x − ( βy + αy ) ˙ x (cid:1) == ( x β ˙ y + x α ˙ y )( x + x ) + ˙ F ′′ ( x + x ) + G (15 x ˙ x + 12 x ˙ x ) + ˙ Hx γ + dη, which can be rewritten as˙ F ′′ ( x + x ) + G (15 x ˙ x + 12 x ˙ x ) + ˙ Hx γ − x (cid:0) ( βy + αy ) ˙ x − ( βy + αy ) ˙ x (cid:1) == − x x β ˙ y − x x α ˙ y − x x β ˙ y − x x α ˙ y − dη Now use the formulas (5) to get rid of the terms ˙ y j at the expense of exact terms, forexample x x β ˙ y = d ( x y β ˙ y ) − x y βd ˙ y . Hence˙ F ′′ ( x + x ) + G (15 x ˙ x + 12 x ˙ x ) + ˙ Hx γ − x (cid:0) ( βy + αy ) ˙ x − ( βy + αy ) ˙ x (cid:1) == x y βd ˙ y + x y αd ˙ y + x y βd ˙ y + x y αd ˙ y + d ˜ η (8)which is an equation in ˙ A = A ⊗ h ˙ x , ˙ x i . Using the formulas for d ˙ y , d ˙ y , d ˙ y andseparating the components in ˙ x , ˙ x independently, we get two equations: F ′′ ( x + x ) + 15 Gx + H x γ − x ( βy + αy ) + d ˜ η == 3 x x y β + 2 x x y α + 2 x x y β + x y α ,F ′′ ( x + x ) + 12 Gx + H x γ + x ( βy + αy ) + d ˜ η == x y β + 2 x x y α + 2 x x y β + 3 x x y α , in the original dga A = Λ( x i ) ⊗ Λ ( y j ). N STRONGLY INFLEXIBLE MANIFOLDS 15 We look at the first equation modulo γ, x . By Lemma 5.8, G = eγ for some e ∈ Q ,and by Lemma 5.6 and Remark 5.7, d ˜ η is in the ideal generated by γ . Hence, the firstequation reduces to F ′′ ( x ) − x ( βy + αy ) = 3 x x y β + 2 x x y α + 2 x x y β. (9)From Lemma 5.8 we know that the ˙ x -part of ˙ F ′′ is F ′′ = x y y + x y y . Then, the lefthand side of (9) is of the form 2 x x y y + x y y . The right hand side of (9) modulo γ, x reduces to − x x y y + 5 x x y y . Hence, we obtain that x y y + 5 x x y y − x x y y = 0 mod ( γ, x ) , which by Lemma 5.9 leads to a contradiction.Therefore, the element X is not in the ideal I . This discussion and the last item inTheorem 4.7 finish our proof. (cid:3) Example 5.3. (Inflexible dgas in literature) We list the collection of inflexible dgas ex-isting in literature that follow the same pattern as the Arkowitz-Lupton example. [2, Ex. 5.1] (cf. [9, Ex. I.3]) [2, Ex. 5.2] (cf. [9, Ex. I.4]) [9, Ex. I.1] dy x x x x x x dy x x x x x x dy x x x x x x dz x αβ + x + x x αβ + x + x x αβ + x + x ν x x x [9, Ex. I.2] [6, Def. 1.1] k ≥ dy x x x x x x dy x x x x x x dy x x x x x x dz x αβ + x + x x k − αβ + x k +51 + x k +42 x αβ + x + x ν x x k +161 ¯ x where x = ¯ x The dgas in these tables are all minimal models of simply-connected inflexible manifolds.In Theorem 5.2 we have carried out in detail the proof that the dga from Definition 5.1(first example of the table) is non-strongly inflexible. The other dgas can be treated byanalogous arguments and calculations as those from Theorem 5.2 to obtain: Theorem 5.4. Let M be one of the inflexible dgas from Example 5.3. Then, M is non-strongly inflexible. Furthermore, any manifold for which M is a Sullivan model is alsonon-strongly inflexible. Remark . The dga in [1, Section 3] is slightly different, since the volume form ν containsa Massey product like element. The dga is (Λ( x , x , y , y , y , y , z ) , d ) with | x | = 4 dx = 0 | x | = 6 dx = 0 | y | = 27 dy = x x | y | = 29 dy = x x | y | = 31 dy = x x | y | = 75 + 4 k dy = x k | z | = 77 dz = x x αβ + x x + x where k ≥ 0, and α = x y − x y , β = x y − x y , δ = x y − x k y are non-trivialMassey products. The volume form is given by ν = x y − x k x y = x δ. We expect that the methods used in this section prove that this dga is non-stronglyinflexible. However, showing that the corresponding map (3) satisfies H ( ψ )([ ν )]) = 0,needs significantly more calculations and our attempts have not proven fruitful yet.As we have previously mentioned, the end of this section is devoted to prove sometechnical lemmas that are needed in the proof of Theorem 5.2. Lemma 5.6. d ˜ η is in the ideal generated by γ. Proof. From (8), we know that d ˜ η ∈ A ⊗ ( Q ⊕ ˙ V ) and that has y -length two. Observethat since ˙ x i ˙ y j = 0, we can express˜ η = X ˜ η i ˙ x i + X ˆ η j ˙ y j where ˜ η i ∈ A ⊗ ( Q ⊕ ˙ V ) and ˆ η j ∈ A ⊗ ( Q ⊕ ˙ V ) . If we apply the differential d ˜ η = X d ˜ η i ˙ x i + X d ˆ η j ˙ y j ± X ˆ η j d ˙ y j we conclude that the elements d ˆ η j = 0 (since d ˜ η does not contain terms with ˙ y j ) andthey are of y -length one. Hence, ˆ η j are closed elements of y -length two. Reasoning withdegrees, we obtain that the only possibilities are:ˆ η = ( a x x + b x x ) γ, ˆ η = ( a x x + b x x ) γ, ˆ η = ( a x x + b x x ) γ, for a i , b i ∈ Q .On the other hand, since the differential decreases by one the y -length, ˜ η is of lengththree and so ˜ η i is, which implies that ˜ η i = R i ( x , x ) y y y . Hence d ˜ η i = R i ( x , x ) d ( y y y ) = R i ( x , x ) x x γ. Gathering all this information, we deduce that d ˜ η is in the ideal generated by γ. (cid:3) Remark . Observe that the ˙ x i -part of d ˜ η is also in the ideal generated by γ , i = 1 , . Lemma 5.8. Let ˙ F ′′ and G be the terms in (7) . Then G = eγ , for some e ∈ Q , and ˙ F ′′ = x ˙ x y y + ( x ˙ x + 2 x ˙ x ) y y + x ˙ x y y . N STRONGLY INFLEXIBLE MANIFOLDS 17 Proof. Recall that ˙ F ′′ = P F ′′ ijk ˙ x k y i y j and that | ˙ F ′′ | = 88 . By degrees reasoning, we get: F ′′ = 0 , F ′′ = ax , F ′′ = bx , F ′′ = cx , F ′′ = px , F ′′ = 0 , a, b, c, p ∈ Q , so˙ F ′′ = ax ˙ x y y + ( bx ˙ x + cx ˙ x ) y y + px ˙ x y y . Applying the differential and using (5), we obtain that d ˙ F ′′ = ˙ x (cid:0) − bx x y − px x y + ( bx x + px x ) y (cid:1) + ˙ x (cid:0) ( − ax x − cx x ) y + ax x y + cx x y (cid:1) We proceed the same way for G = P G ij y i y j and | G | = 88 . By degrees reasoning, weget G = ex , G = f x x and G = gx , e, f, g ∈ Q , so G = ex y y + f x x y y + gx y y . Applying the differential, we get that dG = y ( − e − f ) x x + y ( e − g ) x x + y ( f + g ) x x . We now apply the differential to (8) and compare the ˙ x i components of the new equationfor i = 1 , 2. Denote d ˙ F ′′| ˙ x i the component of d ˙ F ′′ in ˙ x i , for i = 1 , 2. We start by comparingthe ˙ x -components: d ˙ F ′′| ˙ x ( x + x ) + 15 dGx = y ( − x x − x x ) + y ( − x x − x x )+ y (2 x x + 2 x x ) , which is an equation in A . Then, using our computation above of d ˙ F ′′ and dG , we aregoing to compare the y i -components of this last equation, for i = 1 , , . Comparing the y -terms, we get that: − bx x − bx x − f + e ) x x = − x x − x x , so we deduce that b = 1 and that e = − f. Comparing the terms in y , we get that p = 1and e = g .We now compare the ˙ x -components of the equation above: d ˙ F ′′| ˙ x ( x + x ) + dG x = y ( − x x − x x ) + y ( x x − x x )+ y (2 x x + 2 x x ) . Again using our computations above of d ˙ F ′′ and dG and comparing the y i -terms of thislast equation, for i = 1 , , , we obtain that a = 1, and c = 2, which concludes ourproof. (cid:3) Lemma 5.9. The element Z = x y y + 5 x x y y − x x y y is not in the idealgenerated by γ and x . Proof. Let us suppose that Z ∈ ( γ, x ). Since | Z | = 200, by degree reasoning Z can onlybe expressed as follows: Z =( a x + a x + a x x + a x y y + a x y y ) x + ( b x + b x x + b x x ) γ By comparing both sides of the equation, it is clear that a = a = a = 0 . Therefore x y y + 5 x x y y − x x y y = b x y y − b x x y y + b x x y y + b x x y y − b x x y y + b x x y y + b x x y y − b x x y y + b x x y y . From this equation, we get that on the one hand b = 1 and, on the other hand that b = − 5, which is a contradiction. (cid:3) The Costoya-Viruel Example In [8], the authors construct, for any given finite group Γ, a simply-connected ellipticinflexible manifold whose group of self homotopy equivalences is Γ. To that end, theystart with the inflexible dga from Definition 5.1 and add generators corresponding to acertain graph G = ( V, E ), with set of vertices V and set of edges E . These new generatorsinterrelate in such a way to ensure that the self homotopy equivalences of the dga are theautomorphisms of the graph, which happens to be Γ.To analyze the dga of [8], we need the following result on fibrations. Lemma 6.1. Given a commutative diagram of dgas (Λ V , d ) (cid:31) (cid:127) / / ψ (cid:15) (cid:15) (Λ V , d ) / / / / ψ (cid:15) (cid:15) (Λ V , d ) ψ (Λ e V , ˜ d ) (cid:31) (cid:127) / / (Λ e V , e d ) / / / / (Λ V , d ) (10) where each (Λ V i , d i ) , i = 1 , , , has Poincar´e duality cohomology of formal dimension N i ,and volume form η i . If V = V ⊕ V , and e V = e V ⊕ V , then H ( ψ )( η ) = 0 if and onlyif H ( ψ )( η ) = 0 .Proof. By [10, Section 15(a)], since V = V ⊕ V , and e V = e V ⊕ V , the diagram (10) isa Sullivan model for a commutative diagram of rational spaces X X o o X o o e X O O e X o o Ψ O O X o o (11)where the rows are rational fibrations. We proceed by analyzing the rational cohomologySerre spectral sequence (Sss) associated to each fibration in (11), and compare them viathe induced maps connecting both rows.The Sss associated to the top row is E p,q = H p (cid:0) X ; H q ( X ; Q ) (cid:1) = H p ( X ; Q ) ⊗ H q ( X ; Q ) = ⇒ H p + q ( X ; Q ) . Since the rational cohomology of X i is concentrated in degrees at most N i , i = 1 , 3, thenthe group of highest total degree in the E -term is E N ,N = Q h η ⊗ η i , and the class N STRONGLY INFLEXIBLE MANIFOLDS 19 η ⊗ η survives in the E ∞ -term. Therefore, N = N + N and η ⊗ η represents anontrivial multiple of η in the associated graded vector space given by the Sss.We now consider e E p,q , the Sss associated to the bottom row in diagram (11). By meansof the edge morphisms, we also know H ( ψ )( η ) ⊗ H ( ψ )( η ) = H ( ψ )( η ) ⊗ η representsa non trivial multiple of H ( ψ )( η ) in the associated graded vector space given by the e E ∞ -term. Thus if H ( ψ )( η ) = 0, then H ( ψ )( η ) ⊗ η = 0 and H ( ψ )( η ) = 0.Assume now H ( ψ )( η ) = 0, but H ( ψ )( η ) = 0. Then H ( ψ )( η ) ⊗ η represents thezero class in the e E ∞ -term. Since e E p,qn = 0 for q > N , H ( ψ )( η ) ⊗ η can only be trivialin the e E ∞ -term if there exists n ∈ N such that e d n (cid:0) H ( ψ )( η ) ⊗ η (cid:1) = 0 in e E N − n − ,N − nn .However, e d n (cid:0) H ( ψ )( η ) ⊗ η (cid:1) = ( H ( ψ ) ⊗ H ( ψ )) (cid:0) d n ( η ⊗ η ) (cid:1) (by naturality of edge morphisms)= ( H ( ψ ) ⊗ H ( ψ ))(0) (since 0 = η ⊗ η ∈ E N ,N ∞ )= 0 , which is a contradiction. Therefore if H ( ψ )( η ) = 0, then H ( ψ )( η ) = 0. (cid:3) The dga of [8] is defined as follows. Definition 6.2. For a finite and simple connected graph G = ( V, E ) with more than onevertex, we define the minimal Sullivan algebra associated to G as M G = (cid:16) Λ( x , x , y , y , y , z ) ⊗ Λ( x v , z v | v ∈ V ) , d (cid:17) , where degrees and differential are described by | x | = 8 , dx = 0 | x | = 10 , dx = 0 | y | = 33 , dy = x x | y | = 35 , dy = x x | y | = 37 , dy = x x | x v | = 40 , dx v = 0 | z | = 119 , dz = y y x x − y y x x + y y x + x + x | z v | = 119 , dz v = x v + X ( v,w ) ∈ E x v x w x . Theorem 6.3. M G is an elliptic inflexible dga of formal dimension N = 208 + 80 | V | and volume form ν = x Q v ∈ V x v .Proof. This is included in [8] and [9, Proposition I.6]. (cid:3) Theorem 6.4. Let M G be the minimal model in Definition 6.2. Then there exist a dgamorphism ψ : M G → ¯ B such that ¯ B is a finite type dga that has a positive weight and H ( ψ )([ ν ]) = 0 . Therefore M G is non-strongly inflexible.Proof. Let A be the 2-step algebra A = (Λ( x , x , y , y , y , x v , z v ) , d ) ⊂ M G , and define P = x + x , P = y y x x − y y x x + y y x . The algebra A has apositive weight ω given by Proposition 2.13, and following the arguments of Section 4,there exists a dga B := ( ˙ A ⊗ Λ( u , u , u ) , d ), where d ( u ) = ω ( P + ˙ P ) , d ( u ) = ω ( P ) , d ( u ) = ω ( ˙ P ) . such that ω extends to a positive weight on B given by ω ( u ) = ω ( P + ˙ P ) , ω ( u ) = ω ( P ) , ω ( u ) = ω ( ˙ P ) . Let I ⊂ B be the differential ideal generated by the ω -homogeneous elements ˙ x v , ˙ z v ∈ B .Then, according to Lemma 3.4, the dga ¯ B = B / I admits a positive weight e ω . Define thedga map ψ : M G −→ ¯ B ,a ∈ A 7→ ¯ a + ¯˙ a,z ¯ u + ¯ u + ¯ u , (12)and consider the commutative diagram(Λ( x , x , y , y , y , z ) , d (cid:1) (cid:31) (cid:127) / / ψ (cid:15) (cid:15) M G / / / / ψ (cid:15) (cid:15) (cid:0) Λ( x v , z v | v ∈ V ) , d (cid:1) ψ ( B , ˜ d ) (cid:31) (cid:127) / / ¯ B / / / / (cid:0) Λ( x v , z v | v ∈ V ) , d (cid:1) , (13)where ψ : (Λ( x , x , y , y , y , z ) , d (cid:1) → ( B , ˜ d ) is the morphism constructed in (3).Finally, notice that d ( x v ) = 0, and d ( z v ) = x v , thus (cid:0) Λ( x v , z v | v ∈ V ) , d (cid:1) is elliptic.By Proposition 4.6, ψ maps non trivially the volume form of (Λ( x , x , y , y , y , z ). Then H ( ψ )( ν ) = 0 by Lemma 6.1, and M G is non-strongly inflexible by Corollary 2.12. (cid:3) Remark . The same arguments and calculations used to prove that the dgas fromDefinition 6.2, [8, Def. 2.1], are non-strongly inflexible, work to show that the dga’s from[6, Definition 2.1] and [7, Definition 4.1] are also non-strongly inflexible. Hence, as acorollary of Theorem 3.5, we get that any simply-connected manifold admitting one ofthose dgas as Sullivan minimal model, is also a non-strongly inflexible manifold.7. Connected sums and strongly inflexibility In this section we deal with more examples of inflexible manifolds that are producedfrom the manifolds of Sections 5 and 6. Using connected sums and products, infinitelymany oriented closed simply connected inflexible manifolds are constructed in [1] and [9].We develop here all the tools to prove that building upon non-strongly inflexible manifoldsproduces non-strongly inflexible manifolds. Proposition 7.1. Suppose that M is a compact oriented manifold which is non-stronglyinflexible, and let M be any other compact oriented manifold. Then M × M is non-strongly inflexible.Proof. As M is non-strongly inflexible, there exists a compact oriented manifold M ′ suchthat deg( M ′ , M ) is unbounded. Lef λ > f : M ′ → M such that | deg( f ) | ≥ λ .Then f × Id : M ′ × M → M × M has deg( f × Id) = deg( f ), hence | deg( f × Id) | ≥ λ . Sodeg( M ′ × M , M × M ) is unbounded, and hence M × M is non-strongly inflexible. (cid:3) N STRONGLY INFLEXIBLE MANIFOLDS 21 Corollary 7.2. The manifolds of [1, Thm. 3.4] are inflexible but non-strongly inflexible.Proof. They are of the form Q ≤ l ≤ k M l , where M l are taken from the examples given inSection 5, which are non-strongly inflexible by our previous results. Using Proposition7.1, we conclude. (cid:3) Proposition 7.3. Let M , M , M ′ , M ′ be compact oriented N -manifolds. Then deg( M ′ , M ) ∩ deg( M ′ , M ) ⊆ deg( M ′ M ′ , M M ) . Proof. The result is trivial for N = 1 as the manifolds are S and the degree sets are Z . So we assume N ≥ 2. Let d ∈ deg( M ′ , M ) ∩ deg( M ′ , M ), that is, there are maps f : M ′ → M and f : M ′ → M with deg( f ) = deg( f ) = d . By approximation theory,we can slightly perturb f , f to smooth maps, hence we can assume they are both smooth.We want to homotop them to maps g : M ′ → M and g : M ′ → M so that there are p ∈ M , p ∈ M with g − ( p ) = { p ′ } , g − ( p ) = { p ′ } .We do it for g . Take a regular value p ∈ M so that g − ( p ) = { q , . . . , q l } , and g isa diffeomorphism g : U q j → V p between neighborhoods of q j and p , j = 1 , . . . , l . Takepaths γ j from q to q j . We can arrange that they are embedded and do not intersect(except at the end-point q ). Take the star-shaped graph Γ = S lj =2 γ j ⊂ M ′ . Theinclusion Γ ֒ → M ′ is a cofibration. There is a homotopy of f | Γ : Γ → M to a constantmap (relative to q ). We extend it to a homotopy of f : M ′ → M to a continuous map f ′ : M ′ → M . This map can be arranged to that ( f ′ ) − ( p ) = Γ. If we contract, M ′ / Γis homeomorphic to M ′ . The composition M ′ ∼ = M ′ / Γ f ′ −→ M is a map g : M ′ → M with g − ( p ) a single point p ′ .Now take a small ball B around p , and another ball B ′ ⊂ g − ( B ) around p ′ . Let r : B − { p } → ∂B be the radial projection. Then define g ′ : M ′ − B ′ → M − B sothat it equals g on M ′ − g − ( B ) and it equals r ◦ g on g − ( B ) − B ′ . Then g ′ sends h = g ′ | ∂B ′ : ∂B ′ → ∂B . As g has degree d , then h is a degree map on a ( N − g and obtain g ′ and h . As deg( h ) = deg( h ), the maps h , h arehomotopic, say through a homotopy H : S N − × [0 , → S N − .We define a degree d map F : M ′ M ′ → M M as the map ( M ′ − B ′ ) ∪ ( S N − × [0 , ∪ ( M ′ − B ′ ) → ( M − B ) ∪ ( M − B ) equal to g ′ , H, g ′ on each of the pieces. This completes the result. (cid:3) Proposition 7.4. If M is an oriented compact manifold which is non-strongly inflexible,then k M is non-strongly inflexible for all k ≥ .Proof. Let M ′ be so that deg( M ′ , M ) is unbounded. We apply Proposition 7.3 inductivelyto M, M ′ and k M, k M ′ so thatdeg( M ′ , M ) ∩ deg( k M ′ , k M ) ⊆ deg( k +1 M ′ , k +1 M ) . Hence deg( M ′ , M ) ⊆ deg( k M ′ , k M ) for all k ≥ 2. This proves the result. (cid:3) Corollary 7.5. The manifolds of [9, Ex. II.4] and of [1, Cor. 3.5] are inflexible but non-strongly inflexible. Finally, in [9, Proposition II.13], the authors construct nonspinable simply-connectedinflexible manifolds. To deal with them, we need the following: Proposition 7.6. Let M , M be compact oriented non-strongly inflexible N -manifolds.If there is M ′ such that deg( M ′ , M ) = Z , then M M is non-strongly inflexible.Moreover, if M , M , M ′ , M ′ are compact oriented N -manifolds such that there is some s > with s Z ⊂ deg( M ′ , M ) and deg( M ′ , M ) ∩ s Z is infinite, then M M is non-strongly inflexible.Proof. Let M ′ be a compact oriented manifold such that deg( M ′ , M ) is unbounded.The result follows from Proposition 7.3 as deg( M ′ , M ) = Z , implies deg( M ′ , M ) ⊂ deg( M ′ M , M M ). Thus deg( M ′ M , M M ) is unbounded and M M is non-strongly inflexible.The second assertion is proved in a similar way. (cid:3) Let s > Proposition 7.7. Let ( A , d ) be a dga with Poincar´e duality and volume form [ ν ] ∈ H N ( A , d ) . Suppose that there exists a dga ( A ′ , d ′ ) with a positive weight, and a map ϕ : ( A , d ) → ( A ′ , d ′ ) such that H ( ϕ )([ ν ]) = 0 . Then deg( A , A ′ ) ∩ s Z is unbounded.Let ( M, η ) be a simply-connected N -manifold with model ( A , d ) , and η ⊗ Q ν ] . Thenthere exists a manifold M ′ such that deg( M ′ , M ) ∩ s Z is unbounded.Proof. In the proof of Corollary 2.12 we take maps with deg( f n ) ∈ ts Z , where t is thedenominator of (deg q )(deg ϕ ). For this we take in Lemma 2.8, q n ∈ ts Z .The last assertion follows from Theorem 3.5 that hinges in Theorem 3.2. There | deg( α l ) | = | lλ f k | . So it is enough to take l ∈ ts Z , where t is the denominator of λ f k . (cid:3) All manifolds in Sections 5-6 satisfy the conditions of Proposition 7.7, hence the degreesof maps deg( M ′ , M ) ∩ s Z are unbounded for any s > Corollary 7.8. The manifolds of [9, Prop. II.13] are inflexible but non-strongly inflexible.Proof. The manifolds of [9, Prop. II.13] are of the form M × k × W , where M is one ofthe manifolds in previous sections, and W = S N − ˜ × S is the non-trivial fibre bundle S N − → ˜ W → S .With s = 2, there is some M ′ such that deg( M ′ , M × k ) ∩ Z is unbounded. Nowtaking the pull-back under the degree 2 map g : S → S of the non-trivial bundle S N − → ˜ W → S we get a trivial bundle ˜ W = g ∗ W ∼ = S N − × S . As this has self-mapsof any degree, we have that deg( ˜ W , W ) ⊃ Z . Applying now Proposition 7.6, we get that M × k × W is non-strongly inflexible. (cid:3) For completeness, we will include some results on degrees of maps for connected sums,using dgas. First, given subsets A, B ⊂ Z , define A + B = { a + b | a ∈ A, b ∈ B } ⊂ Z . N STRONGLY INFLEXIBLE MANIFOLDS 23 Lemma 7.9. Let M i , i = 1 , , , be compact oriented N -manifolds. Then deg( M , M ) + deg( M , M ) ⊂ deg( M M , M ) . Proof. Let q : M M → M ∨ M denote the pinching map. Then for any given maps f i : M i → M , the map f : M M → M given by the composition f = ( f ∨ f ) ◦ q verifies deg( f ) = deg( f ) + deg( f ), and the result follows. (cid:3) Note that the containment in Lemma 7.9 can be strict. For instance, let M = M = T be the 2-torus and M be the compact oriented surface of genus 2. Corollary 7.10. Let M be a compact oriented and non-strongly inflexible N -manifold.Then there exist infinitely many compact oriented N -manifolds M ′ such that deg( M ′ , M ) is unbounded.Proof. Since M is non-strongly inflexible, there exists a compact oriented N -manifold M ′ such that deg( M ′ , M ) is unbounded. Then, according to Lemma 7.9 for any N -manifolds W , deg( M ′ , M ) ⊂ deg( W M ′ , M ), and therefore deg( W M ′ , M ) is unbounded too. (cid:3) The following is a straightforward consequence of the previous corollary. Corollary 7.11. Let M be a compact oriented N -manifolds, and let v M be the domination Mfd N -seminorm associated with M (see [9, Definition 7.1] ). If v M is not finite, then thereexist infinitely many compact oriented N -manifolds M ′ such that v M ( M ′ ) = ∞ .Proof. Recall that given a compact oriented N -manifold M ′ , v M ( M ′ ) = sup {| n | | n ∈ deg( M ′ , M ) } , thus the result follows from Corollary 7.10. (cid:3) Definition 7.12. Let A i , i = 1 , , be connected dgas, and let a i ∈ A i elements such that | a | = | a | . The connected sum of the pairs ( A i , a i ) , i = 1 , , is the dga ( A , a ) A , a ) := ( A ⊕ Q A ) /I , where A ⊕ Q A := ( A ⊕ A ) / Q h (1 , − i , and I ⊂ A ⊕ Q A is the differential idealgenerated by a − a . The connected sum of dgas provides a model for the connected sum of manifolds. Theorem 7.13. Let M i , i = 1 , , be simply-connected compact oriented N -manifolds.Let ( M i , [ m i ]) be the associated rational model. Then ( M , m ) M , m ) is a rationalmodel of M M .Proof. If ( M i , [ m i ]) is a rational model of M i , then M ⊕ Q M is a rational model of M ∨ M [11, Example 2.47], and therefore dividing by the ideal generated m − m is arational model of M M [11, Example 3.6]. (cid:3) The connected sum of dgas behaves nicely with positive weights Proposition 7.14. Let A i , i = 1 , , be connected dgas, and let a i ∈ A i be elements suchthat | a | = | a | . If A i , i = 1 , , has a positive weight ω i such that a i is ω i -homogeneous,then ( A , a ) A , a ) admits a positive weight. Proof. Let ω be the positive weight in A ⊕ Q A given by ω ( x ) = ( ω ( a ) ω ( x ) , if x ∈ A is ω -homogeneous, ω ( a ) ω ( x ) , if x ∈ A is ω -homogeneous.Then a − a ∈ A ⊕ Q A is ω -homogeneous, hence d ( a − a ) is so. Therefore since I = ( A ⊕ Q A ) · ( a − a , d ( a − a )) , the differential closed ideal I is generated (as vector space) by ω -homogeneous elements,and according to Lemma 3.4, ω gives rise to a positive weight on ( A , a ) A , a ). (cid:3) Finally, Theorem 7.15. Let ( M i , η i ) be a simply-connected N -manifold with minimal model M i =(Λ V i , d ) , i = 1 , . Write the volume form as η i = [ ν i ] . Assume there exist a dga morphisms ψ i : M i → A i such that A i is a finite type dga that has a positive weight and H ( ψ i )([ ν ]) =0 . Then M M is non-strongly inflexible.Proof. Let ω i be a positive weight on A i , i = 1 , 2. We claim ψ i ( ν i ) may be assumed ω i -homogeneous. As in the proof of Theorem 3.2, decompose ψ i ( ν i ) = P α j into ω i -homogeneous elements, fix e a i ∈ A i with nontrivial α j , ω i -homogeneous complements A i = h e a i i ⊕ W i , and define I i = A ≥ N +1 i ⊕ W i . Then by Lemma 3.4, e A i = A i /I i is a finite type connected dga with positive weight,and the induced map e ψ i : M i → e A i maps ν i to a e ω i -homogeneous element. The algebra (cid:0) e A , e ψ ( ν ) (cid:1) (cid:0) e A , e ψ ( ν ) (cid:1) inherits a positive weight by Proposition 7.14. Therefore theobvious morphism e ψ e ψ : ( M , ν ) M , ν ) → (cid:0) e A , e ψ ( ν ) (cid:1) (cid:0) e A , e ψ ( ν ) (cid:1) is a morphism from the rational model of ( M M , η ) to a finite type dga that has a pos-itive weight, such that [( e ψ e ψ )( η )] = 0. Finally, according to Theorem 3.5, ( M M , η )is non-strongly inflexible. (cid:3) References [1] M. Amann, Degrees of Self-Maps of Simply Connected Manifolds , Int. 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Thom, Quelques propri´et´es globales des vari´et´es diff´erentiables , Comment. Math. Helv. (1954),17–86.[16] D. Sullivan, Infinitesimal computations in topology , Inst. Hautes ´Etudes Sci. Publ. Math. (1977),269–331. CITIC, Departamento de Computaci´on, Universidade da Coru˜na, 15071-A Coru˜na, Spain. Email address : [email protected] Departamento de ´Algebra, Geometr´ıa y Topolog´ıa, Universidad de M´alaga, Campusde Teatinos, s/n, 29071 M´alaga, Spain Email address : [email protected] Departamento de ´Algebra, Geometr´ıa y Topolog´ıa, Universidad de M´alaga, Campusde Teatinos, s/n, 29071 M´alaga, Spain Email address ::