aa r X i v : . [ m a t h . A T ] A p r Hodge decompositions and Poincaréduality models
Pavel Hájek ∗ University of HamburgApril, 2020
We use Hodge decompositions to construct Poincaré duality models andimprove results of Lambrechts & Stanley in the simply-connected case. Themain idea is the construction of a certain oriented extension of the Sullivanminimal model which admits a Hodge decomposition.
Contents A Poincaré
DGA of degree n (=: PDGA) is a non-negatively graded unital CDGA V with finite-dimensional homology H( V ) equipped with an orientation o H : H( V ) → R indegree n such that the induced pairing h h , h i H := o H ( h ∧ h ) for h , h ∈ H( V ) is non-degenerate. Such H( V ) is called a Poincaré duality algebra (=: PD-algebra). A mor-phism of PDGA’s V and V (=: PDGA-morphism) is a DGA-morphism f : V → V ∗ E-mail: [email protected] f ∗ : H( V ) → H( V ) preserves the orientation. A differ-ential Poincaré duality algebra of degree n (=: dPD-algebra) is a finite-dimensionalnon-negatively graded unital CDGA M equipped with an orientation o : M → R in degree n such that the induced pairing h v , v i := o( v ∧ v ) for v , v ∈ M isnon-degenerate. A Poincaré duality model (=: PD-model) of a PDGA V is a dPD-algebra M such that there is a zig-zag of PDGA-quasi-isomorphisms connecting M to V .It was shown in [LS08] that a PD-model of a PDGA V exists provided that H ( V ) ≃ R and H ( V ) = 0. Moreover, [LS08, Theorem 7.1] asserts the following. Let M and M be two PD-models of V and suppose that the following holds:(1) n ≥ ( V ) ≃ R , H ( V ) = H ( V ) = H ( V ) = 0 and(3) M i ≃ R , M i = M i = 0 for i ∈ { , } .Then there is a third PD-model M and DGA quasi-isomorphisms M → M ← M .Such statement can be called the “weak uniqueness” of PD-models in contrast to the“strict uniqueness” of Sullivan minimal models. In order to obtain M , they take aSullivan minimal model W := Λ U → V , which is of finite type due to H ( V ) ≃ R andH ( V ) = 0, and construct a certain extension ˆ W which is quasi-isomorphic to W viathe inclusion W ֒ → ˆ W and oriented such that the canonical projection π Q : ˆ W −→ ˆ W / ˆ W ⊥ =: Q ( ˆ W ) , where ˆ W ⊥ := { w ∈ ˆ W | w ⊥ ˆ W } , is a quasi-isomorphism. It is then easy to see that V ← W → Q ( ˆ W ) =: M is a PD-model of V . The extension ˆ W is constructed from W by adding new generators whichcommute with W and “kill”, in the homological sense, generators of the group H( W ⊥ ),called the “orphans”, which obstruct W → Q ( W ) from being a quasi-isomorphism.In [Fio+19], it was observed that for an oriented DGA A , the vanishing of H( A ⊥ )implies that A admits a Hodge decomposition, i.e., that there exists a complement H of im d in ker d, called the harmonic subspace, and a complement C of ker d in A , calledthe coexact part, such that C ⊥ H ⊕ C . They say that A is of Hodge type if it admitsa Hodge decomposition A = H ⊕ d C ⊕ C . In fact, A is of Hodge type if and only ifH( A ⊥ ) = 0 provided that Q ( A ) is of finite type. Based on this observation, we presenta different construction of an oriented extension ˆ W of W which is aimed at obtaininga Hodge decomposition of ˆ W rather than at H( ˆ W ⊥ ) = 0. The obstruction for theexistence of a Hodge decomposition are elements c ∈ C which are not perpendicularto C and which we call “non-degenerates”. We get rid of them by adding their “exactpartners”, i.e., exact elements d w such that h c, ·i = h d w, ·i holds on C . Then we takethe coexact part generated by c − d w instead.The following is Proposition 4.3 in the text. Theorem 1.1.
Let W be a PDGA of degree n which is connected, simply-connectedand of finite type. Then it admits an oriented extension ˆ W of Hodge type which isconnected, of finite type and retracts onto W . If W is oriented, then we can achieve hat the inclusion ι : W ֒ → ˆ W is orientation preserving. If n ≥ , then we can achievethat ˆ W is simply-connected. Using ˆ W from Theorem 1.1 in the proofs of [LS08] and taking care of the orientationso that we get PDGA-quasi-isomorphisms, we obtain the following improvement of[LS08, Theorem 1.1] and [LS08, Theorem 7.1]. It is a combination of Proposition 5.1,Proposition 5.4 and Remark 5.5 in the text. Theorem 1.2. A PDGA V with H ( V ) ≃ R and H ( V ) = 0 admits a connected andsimply-connected PD -model M in the form Λ U V V Q (Λ U V ) =: M , where Λ U V is an oriented extension of Hodge type of a Sullivan minimal model Λ U → V which is connected, simply-connected and of finite type. If M and M are twoconnected and simply-connected PD -models of V and if H ( V ) = 0 , then there isa connected and simply-connected PD -model M together with injective orientationpreserving PDGA -quasi-isomorphisms M M M . ι ι The assumption H ( V ) = 0 is necessary for the statement to hold for all V . We conjecture that the theorem holds even if H ( V ) = 0 provided that one does notinsist on M being simply-connected.Another notion from [Fio+19] is that of a small subalgebra S H ,C ( V ) for a Hodgedecomposition V = H ⊕ d C ⊕ C . By definition, it is the smallest dg-subalgebrawhich admits a Hodge decomposition with harmonic subspace H . If H ( V ) ≃ R andH ( V ) = 0, then S H ,C ( V ) is of finite type and V ← ֓ S H ,C ( V ) ։ Q ( S H ,C ( V )) providesa canonical PD-model of V . This is particularly useful if M is an oriented closed n -manifold and V its de Rham algebra Ω( M ) oriented by R M : Ω( M ) → R . Givena Riemannian metric g on M , there is a canonical Riemannian Hodge decomposition Ω( M ) = H ⊕ dΩ( M ) ⊕ d ∗ Ω( M ), where d ∗ = ± ∗ d ∗ is the Hodge codifferential and H = { ω ∈ Ω( M ) | d ω = d ∗ ω = 0 } the space of harmonic forms. We denote thecorresponding small subalgebra by S g (Ω( M )). If M is connected and H ( M ) = 0,one obtains a canonical PD-model M g := Q ( S g (Ω( M ))). We do not know how M g depends on g but we give an example of an artificial Hodge decomposition Ω( SU (6)) = H ⊕ d C ⊕ C such that Q ( S H ,C (Ω( SU (6))))
6≃ M g for a given biinvariant Riemannianmetric g on SU (6). We also show that de Rham complex of the connected sum3 C P does not admit a PD-model M with just one direct PDGA-quasi-isomorphismbetween M and Ω( C P ).Our interest in PD-models stems from the study of the IBL ∞ -chain-model of equivari-ant Chas-Sullivan string topology of M proposed in [CFL15] (IBL stands for involutivebi-Lie algebra). The space of cyclic cochains of a dPD-algebra carries a canonical dIBL-structure with the Hochschild differential. To obtain the IBL ∞ -chain-model, one startswith the canonical IBL-structure on cyclic cochains of H dR ( M ) and “corrects it” bytwisting with a Maurer-Cartan element constructed from Feynman integrals associatedto trivalent ribbon graphs with a propagator coming from a version of Chern-Simonstheory on M . The Maurer-Cartan element can be interpreted as an effective actionand the twisting can be seen as a homotopy transfer of an ill-defined dIBL-structureon cyclic cochains of Ω( M ), which is a PDGA but not a dPD-algebra, to a well-definedIBL ∞ -structure on cyclic cochains of H dR ( M ). A detailed proof that this constructiongives a chain model of string topology is being prepared by K. Cieliebak and E. Volkov.The problem of finding a propagator and computing integrals explicitly is notoriouslyhard except for the case of M = S n with n = 2 which was done by the author in [Haj19].Moreover, we proved vanishing results for the Maurer-Cartan element suggesting thatif H ( M ) = 0, then the IBL ∞ -chain model depends only on the PDGA-structureof Ω( M ). In this case, we proposed to consider the canonical dIBL-structure on cycliccochains of the PD-model M g instead. We call this approach the algebraic approach and the approach based on the Chern-Simons Maurer-Cartan element the geometricapproach. We conjectured that if H ( M ) = 0, then the resulting IBL ∞ -chain-modelsfrom the algebraic and the geometric approach are IBL ∞ -homotopy equivalent. Zig-zags of connected and simply-connected PDGA’s might be particularly useful becauseour vanishing results for the Maurer-Cartan element apply.The author has recently become aware of [NW19], where Poincaré duality models arealso used in the context of IBL ∞ -algebras and a proof of the string topology conjecturefrom [CFL15] is proposed. Acknowledgements:
I thank Prof. Dr. Hông Vân Lê for suggesting that the existenceof a Hodge decomposition and the acyclicity of the degenerate subcomplex are equiva-lent and for her interest in my work. I thank Thorsten Hertl for explaining to me howto compute the cohomology ring of a connected sum. I thank Prof. Dr. Janko Latschevand Dr. Evgeny Volkov for reading the text and suggesting improvements. I thankProf. Dr. Kai Cieliebak for his supervision during the writing of my Ph.D. thesis whichthis work was a part of.
We always work over the field R . We work in the category of Z -graded vector spaces V = L i ∈ Z V i . We say that W is a subspace of V if it is a direct sum of subspaces W i ⊂ V i for all i ∈ Z . We say that U is a complement of W in V if it is a direct4um of complements U i of W i in V i for all i ∈ Z . If we consider a vector v ∈ V , weautomatically assume that it is homogenous, i.e., v ∈ V i for some i ∈ Z , and denoteits degree i by deg v . If we consider a map f : V → V , we automatically assumethat it is linear and homogenous of degree deg f , which is defined by requiring thatdeg f ( v ) = deg f + deg v for all v ∈ V . A DGA is an abbreviation for a differentialgraded algebra. It is a graded vector space V together with a differential d : V → V ofdegree 1 and an associative product ∧ : V ⊗ V → V of degree 0 such that the Leibnitzidentity d( v ∧ v ) = d v ∧ v +( − deg v v ∧ d v holds for all v , v ∈ V . A unital DGAis a DGA V together with a distinguished element 1 of degree 0 — the unit — such that1 ∧ v = v ∧ v for all v ∈ V . A CDGA is an abbreviation for a commutative DGA,i.e., we require that v ∧ v = ( − deg v deg v v ∧ v for all v , v ∈ V . We say that aunital DGA V is connected if V = span { } and simply-connected if V = 0. Let ( V, d) be a cochain complex. An orientation in degree n ∈ N is a map o : V → R of degree − n such that o ◦ d = 0 and o = 0. The triple ( V, d , o) is called an orientedcochain complex. An orientation o on V induces a canonical orientation o H on H( V )such that o H ([ v ]) = o( v ) for all v ∈ ker d. An orientation o H on H( V ) can be alwaysextended to an orientation o on V . The extension is unique if d V n = 0. A cyclicstructure of degree n ∈ N on ( V, d) is a bilinear form h· , ·i : V ⊗ V → R of degree − n such that for all v , v ∈ V , the following holds: h v , v i = ( − deg v deg v h v , v i , (1) h d v , v i = ( − v deg v h d v , v i . (2)A cyclic structure h· , ·i on V induces a canonical cyclic structure h· , ·i H on H( V ) suchthat h [ v ] , [ v ] i H = h v , v i for all v , v ∈ ker d. A cyclic structure on a DGA ( V, d , ∧ )is a cyclic structure h· , ·i on ( V, d) such that for all v , v , v ∈ V , the following holds: h v ∧ v , v i = ( − deg v (deg v +deg v ) h v ∧ v , v i . (3)We remark that any bilinear form on a unital DGA which satisfies (2) and (3) satisfiesautomatically (1). Lemma 3.1. (a) A non-trivial cyclic structure h· , ·i on a unital DGA ( V, d , ∧ ) in-duces a canonical orientation o on V such that o( v ) = h v, i for all v ∈ V .(b) An orientation o on a CDGA ( V, d , ∧ ) induces a canonical cyclic structure h· , ·i on ( V, d , ∧ ) such that h v , v i = o( v ∧ v ) for all v , v ∈ V .Proof. (a) For all v ∈ V , it holdso(d v ) = h d v, i = −h d1 , v i = 0 . h· , ·i 6 = 0, there are v , v ∈ V such that h v , v i 6 = 0. Theno( v ∧ v ) = h v ∧ v , i = h ∧ v , v i = h v , v i 6 = 0 . (b) For all v , v ∈ V , it holds h v , v i = o( v ∧ v ) = ( − deg v deg v o( v ∧ v ) = ( − deg v deg v h v , v i and h d v , v i = o(d v ∧ v )= o (cid:0) d( v ∧ v ) − ( − deg v v ∧ d v (cid:1) = ( − v o( v ∧ d v )= ( − v deg v o(d v ∧ v )= ( − v deg v h d v , v i . For all v , v , v ∈ V , it holds h v ∧ v , v i = o( v ∧ v ∧ v )= ( − deg v (deg v +deg v ) o( v ∧ v ∧ v )= ( − deg v (deg v +deg v ) h v ∧ v , v i . This finishes the proof.Let h· , ·i be a cyclic structure on a cochain complex ( V, d). A complement H of d V inker d is called a harmonic subspace and a complement C of ker d in V a coexact part. If C ⊥ C ⊕ H , then V = H ⊕ d C ⊕ C is called a Hodge decomposition of V . We saythat ( V, d , h· , ·i ) is of Hodge type if it admits a Hodge decomposition. In this case, forany harmonic subspace H , there is a coexact part C such that V = H ⊕ d C ⊕ C is aHodge decomposition; see [Fio+19, Remark 2.6]. Lemma 3.2 ([CFL15, Lemma 11.1]) . Let h· , ·i be a cyclic structure on a cochaincomplex ( V, d) . Suppose that V is of finite type, i.e., it holds dim V i < ∞ for all i ∈ Z ,and that h· , ·i is non-degenerate. Then V is of Hodge type. The degenerate subspace is defined by V ⊥ := { v ∈ V | v ⊥ V } . If h· , ·i is a cyclic structure on a DGA ( V, d , ∧ ), then V ⊥ is a dg-ideal. The non-degenerate quotient is defined by Q ( V ) := V /V ⊥ . It is a DGA with a canonical cyclic structure h· , ·i Q such that the canonical projection6 Q : V → Q ( V ) satisfies h π Q ( v ) , π Q ( v ) i Q = h v , v i for all v , v ∈ V . By a longexact sequence argument, π Q is a quasi-isomorphism if and only if V ⊥ is acyclic. Lemma 3.3.
Let h· , ·i be a cyclic structure on a cochain complex ( V, d) . Then thefollowing holds:(a) If V is of Hodge type and h· , ·i H non-degenerate, then V ⊥ is acyclic.(see also [Fio+19, Lemma 2.8])(b) If V ⊥ is acyclic and Q ( V ) of finite type, then V is of Hodge type.Proof. (a) Consider a Hodge decomposition V = H ⊕ d C ⊕ C . Let v ∈ V ⊥ be suchthat d v = 0. If v d C , then [ v ] = 0 in H( V ) and the non-degeneracy of h· , ·i H impliesthat there is an h ∈ H such that h v, h i 6 = 0. This contradicts v ∈ V ⊥ . Therefore, thereis a w ∈ C such that v = d w . For any h ∈ H and c , c ∈ C , we have h h + d c + c , w i = h d c , w i = ±h d w, c i = ±h v, c i = 0 . Therefore, it holds w ∈ V ⊥ .(b) Because V ⊥ ⊂ V is an acyclic subcomplex and we work over R , there existsa complementary subcomplex W ⊂ V (this elementary fact is proven in [Haj19,Lemma 6.1.12]). The restriction of h· , ·i to W is non-degenerate and since W ≃ Q ( V )is of finite type, Lemma 3.2 asserts the existence of a Hodge decomposition W = H ⊕ d C ′ ⊕ C ′ . Let C ′′ ⊂ V ⊥ be a complement of d V ⊥ in V ⊥ . Set C := C ′ ⊕ C ′′ . It iseasy to see that V = H ⊕ d C ⊕ C is a Hodge decomposition.Consider a Hodge decomposition V = H ⊕ d C ⊕ C for a cyclic structure h· , ·i ona cochain complex ( V, d). The standard Hodge homotopy is a map P : V → V ofdegree − P ( v ) := ( − c if v = d c for some c ∈ C, v ∈ H ⊕ C. If π H : V → H denotes the canonical projection, then it holds[ P , d] = P ◦ d + d ◦ P = π H − . (4)Suppose that h· , ·i is a cyclic structure on a DGA ( V, d , ∧ ) and V = H ⊕ d C ⊕ C isa Hodge decomposition. The smallest dg-subalgebra W ⊂ V such that H ⊂ W and P ( W ) ⊂ W is called the small subalgebra and is denoted by S H ,C ( V ). It is easy to seethat S = H ⊕ d S ⊕ P ( S ) is a Hodge decomposition of S := S H ,C ( V ). Lemma 3.4.
Let h· , ·i be a cyclic structure on a DGA ( V, d , ∧ ) . Let V = H ⊕ d C ⊕ C be a Hodge decomposition and S := S H ,C ( V ) the corresponding small subalgebra. Thenthe following holds:(a) The vector space S is generated by Kontsevich-Soibelman-like evaluations ofrooted binary trees with l ≥ leaves labeled with elements of H , internal nodes abeled with ∧ and edges labeled with either or P .(b) If V is unital and H( V ) is connected, simply-connected and of finite type, thenso is S .Proof. (a) Denote by T the set of representatives of isomorphism classes of labeledtrees and by hT i the vector space generated by their evaluations. It holds hT i ⊂ S . Asfor ⊃ , it is enough to check that for any T , T ∈ T , it holds T ∧ T , d T , P ( T ) ∈ hT i .The product T ∧ T corresponds to the evaluation of the tree obtained by grafting T and T at the root, and hence it lies in hT i . As for d T , resp. P ( T ), we imagine d,resp. P being applied to the root of T and use the Leibnitz identity and (4) topropagate them to the leaves. We also use that P ◦ P = 0 and d H = P ( H ) = 0. Theclaim follows.(b) Suppose that H is of finite type and it holds H = span { } and H = 0. Let T ∈ T be a labeled tree with l leaves. Contracting the leaves labeled with 1, we canassume that every harmonic form at a leaf has degree at least 2. Any rooted binarytree with l leaves has 2 l − l edges adjacent to theleaves are labeled with since otherwise the evaluation of T vanishes due to P ( H ) = 0.The least degree is achieved if the remaining edges are labeled with P . Therefore, thetotal degree D satisfies D ≥ l − ( l −
1) = l + 1. It follows that S is connected,simply-connected and of finite type.A differential Poincaré duality algebra (=: dPD-algebra) of degree n ∈ N is a non-negatively graded unital CDGA ( V, d , ∧ ) of finite dimension equipped with an orienta-tion o in degree n such that the induced cyclic structure h v , v i = o( v ∧ v ) is non-degenerate. If d = 0, we call V a Poincaré dualiy algebra (=: PD-algebra). A
Poincaré
DGA (=: PDGA) of degree n ∈ N is a non-negatively graded unital CDGA ( V, d , ∧ )whose homology H( V ) is equipped with an orientation o H making it into a Poincaréduality algebra of degree n . An oriented PDGA is a PDGA V together with an orien-tation o : V → R which induces the given orientation o H on H( V ). A dPD-algebra V is of Hodge type by Lemma 3.2, and it is easy to see that the induced cyclic struc-ture h· , ·i H on H( V ) is non-degenerate. Therefore, a dPD-algebra is canonically anoriented PDGA. Given PDGA’s V and V of the same degree, a PDGA -morphism f : V → V is a DGA-morphism such that the induced map f ∗ : H( V ) → H( V )preserves the pairing. Lemma 3.5.
Let V and V be dPD -algebras of degree n , and let f : V → V be a PDGA -quasi-isomorphism. Then f preserves h· , ·i and is thus injective. roof. For v , v ∈ V with deg v + deg v = n , it holds h f ( v ) , f ( v ) i = h ∧ f ( v ) , f ( v ) i = h f ( v ) ∧ f ( v ) , i = h f ( v ∧ v ) , i = h [ f ( v ∧ v )] , [1] i H ( ∗ )= h f ∗ [ v ∧ v ] , f ∗ [1] i H = h [ v ∧ v ] , [1] i H = h v ∧ v , i ( ∗ )= h v , v i , where [ · ] denotes the cohomology class. The non-degeneracy of h· , ·i and the factthat V and V are non-negatively graded imply V n +11 = V n +12 = 0. Therefore, wehave d f ( v ∧ v ) = d( v ∧ v ) = 0, and ( ∗ ) hold. Let h· , ·i be a cyclic structure on a cochain complex ( V, d). If H is a harmonic subspaceand C a coexact part such that H ⊥ C , then the decomposition V = H ⊕ d C ⊕ C is called H -orthogonal . Suppose that H( V ) is of finite type and the induced cyclicstructure h· , ·i H on H( V ) is non-degenerate. For any harmonic subspace H , there isthe orthogonal projection π ⊥ : V → H such that h v, h i = h π ⊥ ( v ) , h i for all v ∈ V and h ∈ H . Given any coexact part C , we can set C ′ := { c − π ⊥ ( c ) | c ∈ C } (5)and obtain an H -orthogonal decomposition V = H ⊕ d C ′ ⊕ C ′ . Lemma 4.1.
Let h· , ·i be a cyclic structure of degree n on a cochain complex ( V, d) such that V admits an H -orthogonal decomposition V = H ⊕ d C ⊕ C . Denote C ⊥ := { c ∈ C | c ⊥ C } and suppose that there is a complement E of C ⊥ in C and a map ρ : M i ≥⌈ n/ ⌉ E i −→ d C such that for all i ≥ ⌈ n ⌉ , e ∈ E n − i , e ∈ E i and c ∈ C ⊥ n − i , the following holds: h e , ρ ( e ) i = h e , e i , (6) h c, ρ ( e ) i = 0 . (7)9 f n = 2 k for some k ∈ N , we suppose additionally that dim E k < ∞ .Then there is acoexact part ˜ C such that V = H ⊕ d ˜ C ⊕ ˜ C is a Hodge decomposition.Proof. Let κ : C → d C be a map satisfying the following: κ ( v ) := ρ ( v ) if v ∈ E i for i > ⌊ n ⌋ , v ∈ C ⊥ i for i ≥ ⌈ n ⌉ , v ∈ C i for i < ⌈ n ⌉ . The case of v ∈ E k if n = 2 k for some k ∈ N is specified as follows. The restrictionof h· , ·i to E k is non-degenerate. If k is even, then h· , ·i : E k ⊗ E k → R is an innerproduct, and because dim E k < ∞ by the assumption, there is an orthonormal basis η , . . . , η m of E k for some m ∈ N . Set κ ( η i ) := 12 ρ ( η i ) for all i ∈ { , . . . , m } . (8)If k is odd, then h· , ·i : E k ⊗ E k → R is a symplectic form, and there is a symplecticbasis η , θ , . . . , η m , θ m of E k for some m ∈ N . We use the convention h θ i , η j i = δ ij for i , j = 1, . . . , m . Set κ ( η i ) := ρ ( η i ) and κ ( θ i ) := 0 for all i ∈ { , . . . , m } . (9)Let ˜ C := { c − κ ( c ) | c ∈ C } . It is again a coexact part perpendicular to H . Let c , c ∈ C with deg c + deg c = n and deg c ≤ deg c . Write c = c ⊥ + e and c = c ⊥ + e for c ⊥ , c ⊥ ∈ C ⊥ and e , e ∈ E . Then the following holds: h c − κ ( c ) , c − κ ( c ) i = h c , c i − h κ ( c ) , c i − h c , κ ( c ) i = h e , e i − h κ ( e ) , e i − h e , κ ( e ) i | {z } ( ∗ ) − h κ ( e ) , c ⊥ i − h c ⊥ , κ ( e ) i | {z } ( ∗∗ ) . It holds ( ∗∗ ) = 0 because of (7). As for ( ∗ ), if deg c < deg c , then( ∗ ) = h e , e i − h e , κ ( e ) i = h e , e i − h e , ρ ( e ) i = 0by (6). If deg c = deg c = k and k is even, we plug in the orthonormal basis and10se (8) to get the following: e = η i , e = η j : ( ∗ ) = h η i , η j i − h κ ( η i ) , η j i − h η i , κ ( η j ) i = h η i , η j i − h η j , κ ( η i ) i − h η i , κ ( η j ) i = h η i , η j i − h η j , ρ ( η i ) i − h η i , ρ ( η j ) i = h η i , η j i − h η j , η i i − h η i , η j i = 0 . If k is odd, we plug in the symplectic basis and use (9) to get the following: e = η i , e = η j : ( ∗ ) = h η j , κ ( η i ) i − h η i , κ ( η j ) i = h η j , η i i − h η i , η j i = 0 ,e = θ i , e = η j : ( ∗ ) = h θ i , η j i − h θ i , κ ( η j ) i = h θ i , η j i − h θ i , η j i = 0 ,e = θ i , e = θ j : ( ∗ ) = 0 . This shows that ˜ C ⊥ ˜ C . Lemma 4.2.
Let ( V, d , ∧ , o) be an oriented PDGA of degree n ≥ . Suppose that V is connected, of finite type and that the induced cyclic structure h· , ·i H on H( V ) isnon-degenerate.(a) Suppose that for some l > ⌈ n ⌉ , there is an H -orthogonal decomposition V = H ⊕ d C ⊕ C , a complement E of C ⊥ in C and a map ρ : E ⌈ n/ ⌉ ⊕ · · · ⊕ E l − −→ d C for which (6) and (7) hold for all i ∈ {⌈ n ⌉ , . . . , l − } . Then we can tensor V withan acyclic Sullivan DGA with finitely many generators in degrees l − and l andobtain a connected PDGA ( ˆ
V , ˆd , ∧ ) with an orientation ˆo : ˆ V → R extending o over the inclusion V ֒ → ˆ V , an H -orthogonal decomposition ˆ V = ˆ H ⊕ ˆd ˆ C ⊕ ˆ C , acomplement ˆ E of ˆ C ⊥ in ˆ C and a map ˆ ρ : ˆ E ⌈ n/ ⌉ ⊕ · · · ⊕ ˆ E l −→ ˆd ˆ C for which (6) and (7) hold for all i ∈ {⌈ n ⌉ , . . . , l } .(b) If n ≥ and V = H ⊕ d C ⊕ C is an H -orthogonal decomposition satisfying V ∧ C ⊥ n − l ⊂ H n − l +1 ⊕ d C ⊥ n − l ⊕ C n − l +1 , (10) then (a) holds also for l = ⌈ n ⌉ . That means that we can construct ( ˆ V , ˆd , ∧ ) , ˆo , ˆ H , ˆ C , ˆ E and a map ˆ ρ : ˆ E ⌈ n/ ⌉ → ˆd ˆ C such that (6) and (7) hold for i = ⌈ n ⌉ . roof. (a) Set m := dim E l and consider the (non-minimal) Sullivan DGAΛ := Λ( w , . . . , w m , z , . . . , z m ) with deg w i = l − , deg z i = l, d z i = 0 , and d w i = z i for all i ∈ { , . . . , m } . Let ˆ V := V ⊗ Λ . We will denote the internal ⊗ by ∧ . Note that this is the final ˆ V if V = 0. Construction of ˆo : It holdsΛ = ∞ M k =0 Λ k with Λ k := M r,m ≥ r + m = k Λ r w ⊗ Λ m z, (11)where Λ r w and Λ m z are the vector spaces generated by monomials w I = w i . . . w i r and z J = z j . . . z j m for multiindices I = { i , . . . , i r } and J = { j , . . . , j m } , respectively.It holds V = H ⊕ d E ⊕ d C ⊥ | {z } d C ⊕ E ⊕ C ⊥ | {z } C . (12)Consider the direct sum decomposition of ˆ V obtained from (11) and (12) using thedistributivity of ∧ and ⊕ . Let ξ , . . . , ξ m be a basis of E l and ξ , . . . , ξ m its dualbasis in E n − l . Define a map ˆo : ˆ V → R by 0 on the complement of V ⊕ E n − l ∧ Λ z ⊕ d E n − l ∧ Λ w in ˆ V and byˆo( v ) := o( v ) for all v ∈ V, ˆo( ξ i ∧ z j ) := o( ξ i ∧ ξ j ) andˆo(d ξ i ∧ w j ) := ( − deg ξ i +1 o( ξ i ∧ ξ j ) for all i, j ∈ { , . . . , m } . In order to show that ˆo is an orientation, we must check that ˆo = 0 and ˆo ◦ d = 0.The first condition is clear because ˆo restricts to o on V . As for the second condition,denote ˆ V k := V ∧ Λ k for k ≥ . It holds ˆ V = L ∞ k =0 ˆ V k and ˆd ˆ V k ⊂ ˆ V k for all k ≥
0. Thus, ˆd ˆ V = L ∞ k =0 ˆd ˆ V k . By thedefinition of ˆo, it holds ˆd ˆ V = d V ⊂ ker o ⊂ ker ˆo and L ∞ k =2 ˆd ˆ V k ⊂ L ∞ k =2 ˆ V k ⊂ ker ˆo.As for ˆd ˆ V , we writeˆ V = span { v ∧ w j , v ∧ z j | v ∈ V, j = 1 , . . . , m } and computeˆd ˆ V = span { ˆd( v ∧ w j ) , ˆd( v ∧ z j ) | v ∈ V, j = 1 , . . . , m } = span { d v ∧ w j + ( − deg v v ∧ z j | v ∈ V, j = 1 , . . . , m } . (13)12e write v ∈ V n − l as v = h + d c + c ⊥ + P mi =1 α i ξ i for h ∈ H n − l , c ∈ C n − l − , c ⊥ ∈ C ⊥ n − l and α i ∈ R , and compute for every j = 1, . . . , m the following:ˆo(d v ∧ w j ) = ˆo (cid:16) d c ⊥ ∧ w j + m X i =1 α i d ξ i ∧ w j (cid:17) = m X i =1 α i ˆo(d ξ i ∧ w j )= m X i =1 ( − deg ξ i +1 α i ˆo( ξ i ∧ ξ j )= ( − n − l +1 m X i =1 α i ˆo( ξ i ∧ z j )= ( − n − l +1 ˆo (cid:16) ( h + d c + c ⊥ ) ∧ z j + m X i =1 α i ξ i ∧ z j (cid:17) = ( − deg v +1 ˆo( v ∧ z j ) . Consequently, it holds ˆd ˆ V ⊂ ker ˆo. Construction of ˆ H : It holds ¯H(Λ) = 0 , and hence
H( ˆ V ) ≃ H( V ) ⊗ H(Λ) = H( V ) byKünneth’s formula. Because H ⊂ ker ˆd , H ∩ im ˆd = 0 and dim( H i ) = dim H i ( V ) =dim H i ( ˆ V ) for every i ≥ , H is a harmonic subspace. Therefore, ˆ H := H is a harmonic subspace in ˆ V . Construction of ˆ C : First, because the inclusion
V ֒ → ˆ V onto ˆ V is a quasi-isomorphismand H( ˆ V ) ≃ L k ≥ H( ˆ V k ) , it holds H( ˆ V k ) = 0 for all k ≥ . For k ≥ , let ˆ C k ⊂ ˆ V k bean arbitrary complement of ker ˆd ∩ ˆ V k = ˆd ˆ V k in ˆ V k . Set ˆ C := C ⊕ ( V ∧ Λ w ) ′ ⊕ M k ≥ ˆ C k , where ( V ∧ Λ w ) ′ is defined similarly as (5). The fact that ( V ∧ Λ w ) ′ , or equivalently V ∧ Λ w , is a complement of ker ˆd ∩ ˆ V = ˆd ˆ V in ˆ V follows from (13). Clearly, ˆ C ⊥ ˆ H . Degreewise description of ˆ C and ˆ C ⊥ : For all n − l ≤ i ≤ l , it holds ˆ C i = C n − l for i = n − l,C i for n − l < i < l − ,C l − ⊕ Λ w for i = l − ,C l ⊕ ( V ∧ Λ w ) ′ for i = l. (14)13t holds ˆ C ⊥ i = { c ∈ C ⊥ n − l | c ⊥ V ∧ Λ w } for i = n − l,C ⊥ i for n − l < i < l − ,C ⊥ l − + Λ w for i = l − , { c ∈ ˆ C l | c ⊥ C n − l } for i = l. (15) Construction of ˆ E : Because ˆ V i ∧ ˆ V j ⊂ ˆ V i + j and ˆ V k ⊂ ker ˆo for k ≥ , it holds ˆ C k ⊂ ˆ C ⊥ for all k ≥ . It also holds E ∩ ˆ C ⊥ ⊂ E ∩ C ⊥ = 0 . Therefore, there exists acomplement ˆ E of ˆ C ⊥ in ˆ C satisfying E ⊂ ˆ E ⊂ C ⊕ ( V ∧ Λ w ) ′ . Comparing (14) to (15), we see that ˆ E i = E i for n − l < i ≤ l − . As for ˆ E l and ˆ E n − l , suppose that V ∧ Λ w ⊥ C ⊥ n − l . (16)Then ˆ C ⊥ n − l = C ⊥ n − l , and hence ˆ E n − l = E n − l . The restriction of h· , ·i to ˆ E n − l ⊗ ˆ E l is non-degenerate in both variables, and hence it induces an isomorphism ˆ E l ≃ ˆ E n − l .It holds E n − l ≃ E l by the assumptions, and thus ˆ E l = E l for dimensional reasons.Suppose that (16) does not hold. We set ˆ ρ := ρ : ˆ E ⌈ n/ ⌉ ⊕ · · · ⊕ ˆ E l − = E ⌈ n/ ⌉ ⊕ · · · ⊕ E l − → ˆd ˆ C and see that ˆ V , ˆ C , ˆ E and ˆ ρ satisfy (6) and (7) for all i ∈ {⌈ n ⌉ , . . . , l − } .This is the assumption of this lemma, and so we can repeat the process of adding theprocess of adding exact partners to non-degenerates for ˆ V . Because dim ˆ C ⊥ n − l < dim C ⊥ n − l , (17)condition (16) will be satisfied after finitely many steps. Construction of ˆ ρ : We define ˆ ρ : ˆ E ⌈ n ⌉ ⊕· · ·⊕ ˆ E l → ˆd ˆ C by ˆ ρ := ρ on ˆ E ⌈ n ⌉ ⊕· · ·⊕ ˆ E l − = E ⌈ n ⌉ ⊕ · · · ⊕ E l − and by ˆ ρ ( ξ i ) := z i for all i = 1 , . . . , m. The validity of (6) and (7) for i ∈ {⌈ n ⌉ , . . . , l } follows from the construction.(b) The proof of (a) works until (14) and (15). For n = 2 l − , it holds ˆ C l − = ˆ C n − l = C n − l ⊕ Λ w, ˆ C l = C l ⊕ ( V ∧ Λ w ) ′ ⊕ (Λ w ) l , and for n = 2 l , we have ˆ C l = ˆ C n − l = C n − l ⊕ ( V ∧ Λ w ) ′ ⊕ (Λ w ) l . Note that (Λ w ) l = 0 unless l = 2 and n ∈ { , } . We can not achieve (16) via repeatedextension because (17) must not hold. Instead, (16) is implied by the assumption (10)14nd the construction of ˆo . For n = 2 l − , we have ˆ C ⊥ n − l = C l − ⊕ Λ w, and for n = 2 l , it holds ˆ C ⊥ n − l = C ⊥ n − l ⊕ ( V ∧ Λ w ) ′ ⊕ (Λ w ) l . The proof of (a) goes on because ˆ E n − l = E n − l .Notice that the problem with n = 0 in (a) and with n = 1 , in (b) is that ⌈ n ⌉ = 1 ,and hence w i would land in degree .We say that a PDGA ˆ V is an extension of a PDGA V if there is an injective PDGA -quasi-isomorphism ι : V ֒ → ˆ V . We say that an extension ˆ V retracts onto V if there isa PDGA -quasi-isomorphism p : ˆ V → V such that p ◦ ι = . Proposition 4.3.
Let V be a PDGA of degree n which is connected, simply-connectedand of finite type. Then it admits an oriented extension ˆ V of Hodge type which isconnected, of finite type and retracts onto V . If V is oriented, then we can achievethat the inclusion ι : V ֒ → ˆ V is orientation preserving. If n ≥ , then we can achievethat ˆ V is simply-connected.Proof. For n = 0 , any connected PDGA is of Hodge type. For n = 1 , there is no simply-connected PDGA . For n = 2 , if V = H ⊕ d C ⊕ C is any H -orthogonal decomposition,then V = span { } , V = 0 , V = H ⊕ C by the assumptions, and hence it isa Hodge decomposition. Suppose that n ≥ . Pick an H -orthogonal decomposition V = H ⊕ d C ⊕ C and apply (b) of Lemma 4.2 to get ˆ V , ˆ H , ˆ C , ˆ E and ˆ ρ : ˆ E ⌈ n/ ⌉ → ˆd ˆ C .Condition (10) is trivially satisfied because V = 0 . Notice that for n ∈ { , } , w i areadded in degree . A recursive application of (a) of Lemma 4.2 for l = ⌈ n ⌉ + 1 , . . . , n gives ˆ V , ˆ H , ˆ C , ˆ E and ˆ ρ : ˆ E ⌈ n/ ⌉ ⊕ · · · ⊕ ˆ E n → ˆd ˆ C . Lemma 4.1 implies that ˆ V isof Hodge type. Because ˆ V arose from V by repeated tensoring with acyclic Sullivan DGA ’s, it retracts onto V . We say that a dPD -algebra M is a Poincaré duality model ( =: PD -model) of a PDGA V if there are PDGA ’s Z , . . . , Z k for some k ∈ N and a zig-zag of PDGA -quasi-isomorphisms V ←− Z −→ Z ←− · · · −→ Z k − ←− Z k −→ M . Such zig-zag is called a weak homotopy equivalence of PDGA ’s. Let ( V, d , ∧ , o) be anoriented PDGA of Hodge type such that H ( V ) = span { } and H ( V ) = 0 . Given aHodge decomposition V = H ⊕ d C ⊕ C , the small subalgebra S H ,C ( V ) is connected,15imply-connected and of finite type by Lemma 3.4. Because S H ,C ( V ) is of Hodgetype, Lemma 3.3 implies that the canonical projection S H ,C ( V ) → Q ( S H ,C ( V )) is aquasi-isomorphism. Therefore, we obtain the following PD -model of V : S H ,C ( V ) V Q ( S H ,C ( V )) =: M , (18)The following is our version of [LS08, Theorem 1.1]. Proposition 5.1. A PDGA V which satisfies H ( V ) = span { } and H ( V ) = 0 admits a connected and simply-connected PD -model M in the form Λ U V V Q (Λ U V ) =: M , (19) where Λ U V is an oriented extension of Hodge type of a Sullivan minimal model Λ U → V which is connected, simply-connected and of finite type.Proof. For n = 0 , the inclusion span { } ֒ → V is a PD -model. For n = 1 , no PDGA satisfies the assumptions. For n ≥ , a Sullivan minimal model Λ U → V exists, isconnected, simply-connected, of finite type, and it holds d(Λ U ) = 0 ; see [FOT08,Theorem 2.24]. Its homology inherits an orientation from H( V ) , and we extend itarbitrarily to Λ U . If we denote W := Λ U , then the following holds for an H -orthogonaldecomposition W = H ⊕ d C ⊕ C for n ∈ { , , } : n = 2 : W = H W = 0 W = span { } n = 3 : W = H W = 0 W = 0 W = span { } n = 4 : W = H ⊕ d C ⊕ C W = C W = H W = 0 W = span { } . Therefore, W = H ⊕ d C ⊕ C is automatically a Hodge decomposition, and thus wecan take d Λ U = Λ U . For n ≥ , apply Proposition 4.3 to Λ U to obtain an orientedextension of Hodge type d Λ U which is connected, simply-connected, of finite type andretracts onto Λ U . Lemma 3.3 asserts that the canonical projection d Λ U → Q ( d Λ U ) isan orientation preserving quasi-isomorphism.The next remark shows that the zig-zag (19) is optimal in the sense that it is notalways possible to shorten it. 16 emark . The de Rham algebra (Ω , d , ∧ ) of the oriented simply-connected closed -manifold C P , where denotes the connected sum, oriented by the integration R C P : Ω → R does not admit a PD -model M with just one arrow h : M → Ω or h ′ : Ω → M . The computation in the proof of Lemma 3.5 shows that h and h ′ must preserve the cyclic structure and thus be injective. This is not possible for h ′ as dim Ω = ∞ and dim M < ∞ . As for h , we can assume that M ⊂ Ω . Let H ⊂ M be an arbitrary harmonic subspace in M . The cohomology ring H(Ω) canbe computed with the help of the Mayer-Vietoris sequence. One obtains
H(Ω) =span { } ⊕ span { k , . . . , k } ⊕ span { v } , where deg v = 4 , deg k i = 2 and it holds k i ∧ k j = 0 if i = j and k i ∧ k i = ± v for all i , j ∈ { , . . . , } . The sign depends onthe orientation of the corresponding C P -factor. We normalize h , v i = 1 . Because H ≃
H(Ω) , it follows that H = span { } ⊕ span { k , . . . , k } ⊕ span { v } for some closed v ∈ Ω and k i ∈ Ω such that [v] = v and [k i ] = k i for all i ∈ { , . . . , } .In general, it holds k i ∧ k j = d η ij and k i ∧ k i = v + d ξ i for some η ij , ξ i ∈ M . If f ∈ C ∞ ( C P ) is not a constant, then f k ( k ∈ N ) arelinearly independent over R , and because dim M < ∞ , it must hold M = span { } .By Poincaré duality, it holds M = span { V } . Therefore, for all p ∈ C P , thefollowing relations must hold: k i ( p ) ∧ k j ( p ) = 0 , k i ( p ) ∧ k i ( p ) = ± v( p ) . Given λ i ∈ R for i ∈ { , . . . , } , then (cid:16) X i =0 λ i k i ( p ) (cid:17) ∧ k j ( p ) = ± λ j v( p ) . Therefore, v( p ) = 0 implies that k ( p ) , . . . , k ( p ) are linearly independent. However,this is impossible because dim(Λ T ∗ p C P ) = (cid:18) (cid:19) = 6 < . This finishes the argument. ⊳ The next remark shows that two PD -models of the same PDGA must not be isomorphiceven if they arise as in (18).
Remark . We will show that the de Rham algebra (Ω , d , ∧ ) of the compact simply-connected -dimensional Lie group SU (6) oriented by the integration R SU (6) : Ω → R PD -models in the form Q ( S ) and Q ( S ) , where S and S aresmall subalgebras corresponding to different Hodge decompositions. The cohomologyring H(Ω) is freely generated by single elements in degrees , , . . . , ; see [MT91,Corollary 3.11]. There exist a biinvariant Riemannian metric g and biinvariant dif-ferential forms η , η , . . . , η ∈ Ω in the corresponding degrees such that the freealgebra H := Λ( η , . . . , η ) is precisely the subspace of harmonic forms { ω ∈ Ω | d ω = d ∗ ω = 0 } for the metric g ;see [FOT08, Chapter 1]. For ξ ∈ Ω and ξ ∈ Ω , which are going to be specifiedlater, consider η ′ := η + d ξ and η ′ := η + d ξ . Let H be the vector space obtained from H by replacing η and η with η ′ and η ′ ,respectively. We emphasize that we replace just the vectors and not their products;e.g., η ∧ η is an element of H but η ′ ∧ η ′ might not be. The small subalgebra S of theRiemannian Hodge decomposition Ω = H ⊕ dΩ ⊕ d ∗ Ω clearly satisfies S = H . Let S be the small subalgebra corresponding to a Hodge decomposition V = H ⊕ d C ⊕ C which exists by [Fio+19, Remark 2.6]. Lemma 3.4 implies that the following elementsin degrees , resp. must be contained in S ( P denotes the standard Hodgehomotopy for the second Hodge decomposition): y := P ( η ′ ∧ η ′ − η ∧ η ) = P (cid:0) d( ξ ∧ η − η ∧ ξ + ξ ∧ d ξ ) (cid:1) ,z := η ′ ∧ η − η ∧ η = d( ξ ∧ η ) . Using Stokes’ theorem and d ◦ P = − π im d , we get h y, z i = Z SU (6) P (cid:0) d( ξ ∧ η ) (cid:1) ∧ d( ξ ∧ η )= − Z SU (6) d( ξ ∧ η − η ∧ ξ + ξ ∧ d ξ ) ∧ ξ ∧ η = − Z SU (6) d ξ ∧ η ∧ ξ ∧ η − Z SU (6)
12 d (cid:0) ( η + d ξ ) ∧ ξ ∧ ξ ∧ η (cid:1) = − Z SU (6) d ξ ∧ ξ ∧ η ∧ η . We claim that the integral can be made non-zero by a choice of ξ and ξ . Because [ η ∧ η ] = 0 in H(Ω) , there is a p ∈ SU (6) such that η ( p ) ∧ η ( p ) = 0 . Picklocal coordinates ( x i ) around p such that x i ( p ) = 0 for all i ∈ { , . . . , } . Write η ( p ) ∧ η ( p ) = P I ⊂{ ,..., } , | I | =20 α I dx I and suppose that α I = 0 for some I ⊂{ , . . . , } with | I | = 20 . Consider the complement J of I in { , . . . , } and write J = J ∪ J for some J = { J , . . . , J } and J = { J , . . . , J } . Locally, set ξ := x J dx J \{ J } and ξ := dx J . on a neighborhood of p and ona neighborhood of the complement of the coordinate chart gives globally defined formson SU (6) . By the construction, the integrand d ξ ∧ ξ ∧ η ∧ η is non-zero around p .Because the h· , ·i is non-degenerate, there is a function f ∈ C ∞ ( SU (6)) such that h f, d ξ ∧ ξ ∧ η ∧ η i = Z SU (6) f d ξ ∧ ξ ∧ η ∧ η = 0 . Replacing ξ with f ξ makes h y, z i non-zero. Consider the canonical projection π Q : S → Q ( S ) . Because h y, z i 6 = 0 , the element π Q ( z ) ∈ Q ( S ) is non-zero. Because π Q is a chain map and z exact, π Q ( z ) is exact too. Because π Q is a quasi-isomorphismand η ∧ η generates non-trivial homology, π Q ( η ∧ η ) does that too. It followsthat the vectors π Q ( z ) and π Q ( η ∧ η ) can not be multiples of each other, and hence dim Q ( S ) ≥ . On the other hand, it holds Q ( S ) = H = h η ∧ η i . This showsthat Q ( S ) and Q ( S ) can not be isomorphic. ⊳ Instead of “strict uniqueness”, we have “weak uniqueness” of PD -models. The followingproposition is our version of [LS08, Theorem 7.1]. Proposition 5.4.
Let V and V be connected and simply-connected dPD -algebras ofdegree n which are weakly homotopy equivalent as PDGA ’s. If H ( V ) = H ( V ) = 0 ,then there is a connected and simply-connected dPD -algebra V and injective orienta-tion preserving PDGA -quasi-isomorphisms V V V . h h Proof.
For n = 0 , it holds V = V = span { } . For n = 1 , no simply-connected dPD -algebra exists. For each n ∈ { , , } , it follows from V = V = span { } , V = V = 0 and Poincaré duality that V = H( V ) and V = H( V ) . Given a zig-zag of PDGA -quasi-isomorphisms V ←− Z −→ Z ←− Z −→ Z ←− · · · ←− Z k −→ V , (20)let h : H( V ) → H( V ) be the induced map on homology. Then V := H( V ) , h := h and h := does the job. Suppose that n ≥ and consider a Sullivan minimalmodel Λ U → Z of the DGA Z from the zig-zag (20). Because H ( Z ) ≃ R and H ( Z ) = H ( Z ) = 0 , the inductive construction of the Sullivan minimal model showsthat U = U = 0 . The Lifting Lemma [FOT08, Lemma 2.15] asserts the existenceof DGA -quasi-isomorphisms Λ U → Z and Λ U → Z such that the following diagram19ommutes up to homotopy of DGA ’s: Λ UZ Z Z The diagram commutes strictly at the level of homology, and hence there is an orienta-tion on
H(Λ U ) such that the three vertical arrows become PDGA -quasi-isomorphisms.Hence, we can replace the segment V ←− Z −→ Z ←− Z −→ Z in (20) by theshorter segment V ←− Λ U −→ Z . Repeating this process, (20) can be shortened to Λ UV V , f f where H(Λ U ) is oriented, i.e., Λ U is a PDGA , and f and f are PDGA -quasi-isomorphisms. We will now closely follow the proof of [LS08, Theorem 7.1]. The onlydifference is that we take care of orientations and use the oriented extension of Hodgetype from Proposition 4.3 instead of the extension obtained by killing the orphans.Consider the relative Sullivan minimal model of the multiplication µ : Λ U ⊗ Λ U → Λ U .By [FOT08, Example 2.48], it is given by the commutative diagram Λ U Λ U ⊗ Λ UM =: Λ U ⊗ Λ U ⊗ Λ(s U ) , µ ιp where ι is the inclusion into the first two factors, which is a cofibration, p is a surjec-tive quasi-isomorphism and s denotes the suspension defined by (s U ) i = U i +1 . Forcompleteness, given u ∈ U , the differential D on M satisfies D ( u ⊗ ⊗
1) = d u ⊗ ⊗ , D (1 ⊗ u ⊗
1) = 1 ⊗ d u ⊗ and D (1 ⊗ ⊗ s u ) = u ⊗ ⊗ − ⊗ u ⊗ γ u for a decomposableelement γ u ∈ M . We orient H( M ) such that p becomes a PDGA -quasi-isomorphism.For i ∈ { , } , consider the commutative diagram Λ U V i Λ U ⊗ Λ U V ⊗ V M M ⊗ Λ U ⊗ Λ U ( V ⊗ V ) =: W, f i ι i η i f ⊗ f ι ηf where ι i : Λ U → Λ U ⊗ Λ U and η i : V i → V ⊗ V are the canonical inclusions into the20 -th factor and the lower square with maps f and η is a pushout diagram; see [Men15,Example 1.4]. It is well-known that the model category of non-negatively graded unital CDGA ’s is proper, and hence pushouts along cofibrations preserve quasi-isomorphisms.Therefore, f : M → W is a DGA -quasi-isomorphism. We orient H( W ) such that W becomes a PDGA and f a PDGA -quasi-isomorphism. At the level of algebras, it holds W ≃ V ⊗ V ⊗ Λ(s U ) . From V = V = span { } , V = V = 0 and U = U = 0 it follows that W isconnected and simply-connected. It is also of finite type because V , V and Λ(s U ) are.Proposition 4.3 for n ≥ applies and gives an oriented extension of Hodge type ˆ W which is connected, simply-connected and of finite type. Consider the composition π : W ֒ → ˆ W → Q ( ˆ W ) , where the first map is the inclusion and the second mapthe canonical projection π Q : ˆ W → Q ( ˆ W ) . It is a PDGA -quasi-isomorphism byLemma 3.3. For i ∈ { , } , we define h i := π ◦ η ◦ η i : V i −→ Q ( ˆ W ) =: V and compute using ( η ◦ η i ) ◦ f i = f ◦ ( ι ◦ ι ) and p ◦ ( ι ◦ ι i ) = ( p ◦ ι ) ◦ ι i = µ ◦ ι i = that ( h i ) ∗ = π ∗ ◦ f ∗ ◦ ( ι ◦ ι ) ∗ ◦ ( f i ) − ∗ = π ∗ ◦ f ∗ ◦ p − ∗ ◦ ( f i ) − ∗ . Therefore, h and h are PDGA -quasi-isomorphisms and since V , V , V are dPD -algebras, Lemma 3.5 implies that they are orientation preserving inclusions.The next remark shows that the assumption H ( V ) = H ( V ) = 0 can not be omitted. Remark . We will construct dPD -algebras V and V satisfying all assumptions ofProposition 5.4 except for H ( V ) = H ( V ) = 0 and show that there is no simply-connected dPD -algebra V which admits PDGA -quasi-isomorphisms V → V ← V .Consider the minimal Sullivan DGAΛ := Λ( a, b, c ) with deg a = 2 , deg b = 3 , deg c = 5 , d a = d c = 0 , d b = a . It holds
H(Λ) = span { [1] , [ a ] , [ c ] , [ ac ] } . We set V := H(Λ) and equip it with an orientation o : V → R satisfying o([ ac ]) = 1 .This makes V into a connected and simply-connected dPD -algebra. Consider the21ollowing unital CDGA : V := span { , k, w, z, l, v } with deg k = 2 , deg w = 3 , deg z = 4 , deg l = 5 , deg v = 7 k ∧ l = v, k ∧ k = z, z ∧ w = v, k ∧ w = l, d k = d z = d l = d v = 0 , d w = z. We equip V with an orientation o : V → R satisfying o( v ) = 1 . This makes V intoa connected and simply-connected dPD -algebra. The following morphisms of algebrasare PDGA -quasi-isomorphisms: f : Λ −→ V a [ a ] b c [ c ] f : Λ −→ V a kb wc l. Therefore, V and V are weakly homotopy equivalent as PDGA ’s. Suppose that thereis a dPD -algebra V and PDGA -quasi-isomorphism h : V → V and h : V → V .By Lemma 3.5, h and h are injective and orientation preserving. The images h ([ a ]) and h ( k ) are closed elements in V and because [ a ] ∧ [ a ] = 0 and k ∧ k = z = 0 , theycan not be multiples of each other. Because H ( V ) ≃ R , non-zero elements in V whose differentials kill the additional closed elements in V must exist. ⊳ Conjecture 5.6.
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