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## MASSEY PRODUCT AND TWISTED COHOMOLOGY OF A-INFINITY ALGEBRAS

Citations: | 1 - 0 self |

### Citations

788 |
Homological algebra
- Cartan, Eilenberg
- 1956
(Show Context)
Citation Context .... 30 WEIPING LI AND SIYE WU Appendix. Spectral sequences We summarize some aspects of spectral sequences used in Section 5. We refer the readers to numerous discussions in the literature (for example =-=[4, 23]-=-) although the version we present here is somewhat different. We will treat the cohomological spectral sequences only as the homological ones are parallel. Let (C•, d) be a (Z-graded) cochain complex ... |

454 |
Homotopy associativity of H-spaces
- Stasheff
- 1963
(Show Context)
Citation Context ...converging to the twisted cohomology groups and show that the higher differentials are given by the A∞-algebraic Massey products. 1. Introduction The concept of A∞-algebras was introduced by Stasheff =-=[28, 29]-=- for studying multiplication operations which satisfy associativity up to homotopy. Since then it has played a crucial role in homotopy theory. The A∞-structure on the cohomology of a topological spac... |

251 | Homological mirror symmetry and torus fibrations. Symplectic geometry and mirror symmetry (Seoul
- Kontsevich, Soibelman
(Show Context)
Citation Context ...ational homotopy type of 1-connected spaces [11]. Recently, the subject finds applications in many areas of algebra, topology, geometry and mathematical physics, including homological mirror symmetry =-=[13]-=-. Motivated by the work on twisted cohomology of the de Rham complex [27, 1, 20, 16], we study the twisted cohomology of A∞-algebras. In addition, we define higher Massey products on the cohomology of... |

197 |
A user's guide to spectral sequences
- McCleary
- 2001
(Show Context)
Citation Context ...r ≥ 0; (ii) dr = 0 if r is even; (iii) Ep0̄2m = E p0̄ 2m+1 for any p ∈ Z and m ≥ 1. (iv) Ep0̄2 = E p0̄ 3 = H p(A). Proof: The filtration is clearly exhaustive and weakly convergent. By Theorem 3.2 of =-=[23]-=-, the corresponding spectral sequence converges to the twisted cohomology group. (i) Ep1̄r = 0 for all r ≥ 0 since E p1̄ 0 = Gr pAp+1 = 0. (ii) If r is even, then either q or q − r + 1 is odd. So dr :... |

116 | Introduction to A-infinity Algebras and Modules
- Keller
(Show Context)
Citation Context ...i.e., (A •, ∂), where ∂ = b1, is a cochain complex. For n ≥ 2, the meaning of the identities is best seen from the desuspended version. The A∞-algebras defined here differ from the original ones (see =-=[25, 12]-=- for reviews) by a suspension. Let s : s−1A → A be the suspension map of degree −1 given by the identification (s−1A)p = Ap−1. Then the multilinear maps mn = ± s −1 ◦ bn ◦ s ⊗n : (s−1A)⊗n → s−1A (each... |

96 | Strong homotopy algebras of a Kähler manifold
- Merkulov
- 1999
(Show Context)
Citation Context ...A∞-algebra (A, {bn}), the cohomology H(A) = H(A, ∂) is an associative graded algebra under the operation b̄2 induced by b2. In fact, the cohomology H(A) has an A∞-algebra structure {b̄n} with b̄1 = 0 =-=[7, 8, 24, 13]-=-. There is a quasi-isomorphism of A∞-algebras H(A) → A lifting the identity map of H(A). Such an A∞-structure on H(A) is unique up to isomorphisms of A∞-algebras. We describe the A∞-algebra (H(A), {b̄... |

55 |
Matric massey products
- May
- 1969
(Show Context)
Citation Context ... Corollary 3.2(i) to Mat (m+1) + (A), we get∑ r,t≥0 br+1+t(ās· · ·sā︸ ︷︷ ︸ r timess∂aasas· · ·sa︸ ︷︷ ︸ t times ) = 0. This is the non-associative, A∞-algebraic version of the Bianchi identity. (See =-=[14, 22, 2]-=- for the case of differential graded algebras). Furthermore, by Corollary 3.2(iii), we get ∂ā∂ac = − ∑ r,t≥0 br+t+2(ās· · ·sā︸ ︷︷ ︸ r timess∂aasas· · ·sa︸ ︷︷ ︸ t timessc), where c can be either in ... |

53 |
The algebraic structure in the homology of an A(∞)-algebra
- Kadeishvili
- 1982
(Show Context)
Citation Context ...A∞-algebra (A, {bn}), the cohomology H(A) = H(A, ∂) is an associative graded algebra under the operation b̄2 induced by b2. In fact, the cohomology H(A) has an A∞-algebra structure {b̄n} with b̄1 = 0 =-=[7, 8, 24, 13]-=-. There is a quasi-isomorphism of A∞-algebras H(A) → A lifting the identity map of H(A). Such an A∞-structure on H(A) is unique up to isomorphisms of A∞-algebras. We describe the A∞-algebra (H(A), {b̄... |

53 |
The Antisymmetric Tensor Field in Superstring Theory
- Rohm, Witten
- 1986
(Show Context)
Citation Context ...inds applications in many areas of algebra, topology, geometry and mathematical physics, including homological mirror symmetry [13]. Motivated by the work on twisted cohomology of the de Rham complex =-=[27, 1, 20, 16]-=-, we study the twisted cohomology of A∞-algebras. In addition, we define higher Massey products on the cohomology of a general A∞-algebra with a possibly non-associative multiplication. We then constr... |

51 | DG Coalgebras and formal stacks
- Hinich
(Show Context)
Citation Context ...i1+· · ·+ir+j1+· · ·+jt = n (i1, . . . , ir, j1, . . . , jt > 0) and the second runs over r, t ≥ 0, s ≥ 1, r + s+ t = n. Homotopy is an equivalence relation on the set of morphisms.1 1See for example =-=[25, 6, 15]-=-. We thank B. Keller for providing the references. 4 WEIPING LI AND SIYE WU 2.2. The bar construction. Amore conceptual way of describingA∞-structures is through the use the bar construction [29]. Rec... |

35 |
Massey higher products
- Kraines
- 1966
(Show Context)
Citation Context ...as The triple Massey product [19], was generalized to the context of A∞-algebras [30]. We give an explicit construction of the higher Massey products, usually defined for differential graded algebras =-=[14, 21]-=-, for A∞-algebras. Furthermore, we introduce an equivalence relation on the set of defining systems under which the Massey product takes the same value in the cohomology; this clarifies the dependency... |

24 |
les A∞-catégories. Thèse de doctorat, Université Denis Diderot – Paris 7
- Sur
- 2003
(Show Context)
Citation Context ...i1+· · ·+ir+j1+· · ·+jt = n (i1, . . . , ir, j1, . . . , jt > 0) and the second runs over r, t ≥ 0, s ≥ 1, r + s+ t = n. Homotopy is an equivalence relation on the set of morphisms.1 1See for example =-=[25, 6, 15]-=-. We thank B. Keller for providing the references. 4 WEIPING LI AND SIYE WU 2.2. The bar construction. Amore conceptual way of describingA∞-structures is through the use the bar construction [29]. Rec... |

22 |
Some Higher Order Cohomology Operations, in Symposium internacional de topologia algebraica
- Massey
- 1958
(Show Context)
Citation Context ... . . , h︸ ︷︷ ︸ n times ) = 0 and hn(h, . . . , h︸ ︷︷ ︸ n times ) = 0. Therefore Fh(h) = f1(h) and Hh(h) = h1(h). 4. Higher Massey products on the cohomology of A∞-algebras The triple Massey product =-=[19]-=-, was generalized to the context of A∞-algebras [30]. We give an explicit construction of the higher Massey products, usually defined for differential graded algebras [14, 21], for A∞-algebras. Furthe... |

19 |
Twisted K-theory and cohomology, Inspired by
- Atiyah, Segal
- 2006
(Show Context)
Citation Context ...inds applications in many areas of algebra, topology, geometry and mathematical physics, including homological mirror symmetry [13]. Motivated by the work on twisted cohomology of the de Rham complex =-=[27, 1, 20, 16]-=-, we study the twisted cohomology of A∞-algebras. In addition, we define higher Massey products on the cohomology of a general A∞-algebra with a possibly non-associative multiplication. We then constr... |

18 | Analytic torsion for twisted de Rham complexes, [arXiv:0810.4204
- Mathai, Wu
(Show Context)
Citation Context ...inds applications in many areas of algebra, topology, geometry and mathematical physics, including homological mirror symmetry [13]. Motivated by the work on twisted cohomology of the de Rham complex =-=[27, 1, 20, 16]-=-, we study the twisted cohomology of A∞-algebras. In addition, we define higher Massey products on the cohomology of a general A∞-algebra with a possibly non-associative multiplication. We then constr... |

16 | Capital Structure
- SC
(Show Context)
Citation Context ...br ◦ (qi1 ⊗ · · · ⊗ qir), qn = n∑ r=2 ∑ i1+···+ir=n i1,...,ir>0 h1 ◦ br ◦ (qi1 ⊗ · · · ⊗ qir), respectively, for any n ≥ 2. When A is a differential graded algebra, the inductive formulas simplify to =-=[17]-=- b̄n = n−1∑ i=1 p1 ◦ b2 ◦ (qi ⊗ qn−i), qn = n−1∑ i=1 h1 ◦ b2 ◦ (qi ⊗ qn−i). 6 WEIPING LI AND SIYE WU Finally, if (A, {bn}) is a C∞-algebra, then so is (H(A), {b̄n}) [5]. 3. Twisting elements and twist... |

16 |
Algèbres différentielles fortement homotopiquement associatives
- Prouté
- 1984
(Show Context)
Citation Context ...i.e., (A •, ∂), where ∂ = b1, is a cochain complex. For n ≥ 2, the meaning of the identities is best seen from the desuspended version. The A∞-algebras defined here differ from the original ones (see =-=[25, 12]-=- for reviews) by a suspension. Let s : s−1A → A be the suspension map of degree −1 given by the identification (s−1A)p = Ap−1. Then the multilinear maps mn = ± s −1 ◦ bn ◦ s ⊗n : (s−1A)⊗n → s−1A (each... |

14 |
Transferring homotopy commutative algebraic structures
- Cheng, Getzler
(Show Context)
Citation Context ...ructure {b̄n} on H(A) and the quasi-isomorphism q = {qn} : H(A) → A can be expressed explicitly as a sum over the oriented rooted planar trees [13]. Alternatively, they can be obtained inductively by =-=[5]-=- b̄n = n∑ r=2 ∑ i1+···+ir=n i1,...,ir>0 p1 ◦ br ◦ (qi1 ⊗ · · · ⊗ qir), qn = n∑ r=2 ∑ i1+···+ir=n i1,...,ir>0 h1 ◦ br ◦ (qi1 ⊗ · · · ⊗ qir), respectively, for any n ≥ 2. When A is a differential graded... |

13 | Tǎimanov, Massey products in symplectic manifolds - Babenko, A - 2000 |

7 |
On the homology theory of fibre spaces, Uspekhi Mat. Nauk 35:3
- Kadeishvili
- 1980
(Show Context)
Citation Context ...s played a crucial role in homotopy theory. The A∞-structure on the cohomology of a topological space determines the cohomology of its loop space and can be applied to the cohomology of fiber bundles =-=[7]-=-. Moreover, it determines the rational homotopy type of 1-connected spaces [11]. Recently, the subject finds applications in many areas of algebra, topology, geometry and mathematical physics, includi... |

6 |
Twisted tensor products
- Jr
(Show Context)
Citation Context ...n general, b(τh(h)) = 0 means that τh(h) is a morphism of differential graded coalgebras from k (with the trivial coderivation) to TA. So τh(h) ∈ Hom(k, TA) is a twisted cochain in the sense of Brown =-=[3, 25]-=-. We refer the readers to [31] for a history on twisting elements and twisting cochains, and to the references therein as well as [25, 9]. Theorem 3.4. If h is a twisting element of an A∞-algebra A, t... |

6 |
The cohomology of augmented algebras and generalized Massey products for DGAAlgebras
- May
- 1966
(Show Context)
Citation Context ...as The triple Massey product [19], was generalized to the context of A∞-algebras [30]. We give an explicit construction of the higher Massey products, usually defined for differential graded algebras =-=[14, 21]-=-, for A∞-algebras. Furthermore, we introduce an equivalence relation on the set of defining systems under which the Massey product takes the same value in the cohomology; this clarifies the dependency... |

3 |
H-spaces from a homotopy point of view, Lecture Notes in Mathematics 161
- Stasheff
- 1970
(Show Context)
Citation Context ... ︸ n times ) = 0. Therefore Fh(h) = f1(h) and Hh(h) = h1(h). 4. Higher Massey products on the cohomology of A∞-algebras The triple Massey product [19], was generalized to the context of A∞-algebras =-=[30]-=-. We give an explicit construction of the higher Massey products, usually defined for differential graded algebras [14, 21], for A∞-algebras. Furthermore, we introduce an equivalence relation on the s... |

2 | Cohomology C∞-algebra and rational homotopy type
- Kadeishvili
(Show Context)
Citation Context ... of a topological space determines the cohomology of its loop space and can be applied to the cohomology of fiber bundles [7]. Moreover, it determines the rational homotopy type of 1-connected spaces =-=[11]-=-. Recently, the subject finds applications in many areas of algebra, topology, geometry and mathematical physics, including homological mirror symmetry [13]. Motivated by the work on twisted cohomolog... |

1 |
The predifferential of a twisted product, Uspekhi Mat. Nauk 41
- Kadeishvili
- 1986
(Show Context)
Citation Context ... τh(h) ∈ Hom(k, TA) is a twisted cochain in the sense of Brown [3, 25]. We refer the readers to [31] for a history on twisting elements and twisting cochains, and to the references therein as well as =-=[25, 9]-=-. Theorem 3.4. If h is a twisting element of an A∞-algebra A, then (i) b ◦ τh = τh ◦ ∂h on A; (ii) b preserves the subspace τh(A) ⊂ TA; (iii) ∂2h = 0 on A. Proof: (i) follows from Lemma 3.1 since h̄ =... |

1 | Twisting elements in homotopy G-algebras, preprint (2007), to appear in Festschrift in honor - Kadeishvili |

1 |
On a spectral sequence for twisted cohomologies, preprint
- Li, Liu, et al.
- 2009
(Show Context)
Citation Context |

1 |
Vers un Zp-lemme de Hirsch, in: Homotopie Algébrique et Algèbre Locale
- Prouté
- 1984
(Show Context)
Citation Context ...hat there is an A∞-structure {b̄n} on H(A) after choosing the maps p1 and q1 (§2.4). When A is a differential graded algebra, it was a folklore that the A∞-structure on H(A) gives the Massey products =-=[29, 7, 26, 17]-=-. The precise statement seems to be a long standing puzzle.3 We now establish the exact relationship in the more general context of A∞-algebras. Proposition 4.17. Let A be an A∞-algebra and α1, . . . ... |

1 |
A twisted tale of cochains and connections, preprint (2009), arXiv:0902.4396[math.AT
- Stasheff
(Show Context)
Citation Context ...t τh(h) is a morphism of differential graded coalgebras from k (with the trivial coderivation) to TA. So τh(h) ∈ Hom(k, TA) is a twisted cochain in the sense of Brown [3, 25]. We refer the readers to =-=[31]-=- for a history on twisting elements and twisting cochains, and to the references therein as well as [25, 9]. Theorem 3.4. If h is a twisting element of an A∞-algebra A, then (i) b ◦ τh = τh ◦ ∂h on A;... |