## A cubical model of a fibration

Venue: | J. Pure Appl. Algebra |

Citations: | 8 - 7 self |

### BibTeX

@ARTICLE{Kadeishvili_acubical,

author = {Tornike Kadeishvili and Samson Saneblidze},

title = {A cubical model of a fibration},

journal = {J. Pure Appl. Algebra},

year = {},

pages = {203--228}

}

### OpenURL

### Abstract

Abstract. In the paper the notion of a truncating twisting function from a simplicial set to a cubical set and the corresponding notion of twisted Cartesian product of these sets are introduced. The latter becomes a cubical set. This construction together with the theory of twisted tensor products for homotopy G-algebras allows to obtain multiplicative models for fibrations. 1.

### Citations

272 |
On the structure of Hopf algebras
- Milnor, Moore
(Show Context)
Citation Context ...ebra (dgha) (C, µ, ∆) is simultaneously a connected dga (C, µ) and a connected dgc (C, ∆) such that ∆ : C → C ⊗ C is an algebra map; note that a graded connected Hopf algebra has a canonical antipode =-=[26]-=-, so that the antipode is not an issue. A dga M is a (left) comodule over a dgha C if ν : M → C ⊗ M is a dga map. Let (M ′ , ν ′ ) and (M, ν) be comodules over C ′ and C, respectively, and let ϕ : C ′... |

228 |
The cohomology structure of an associative ring
- Gerstenhaber
(Show Context)
Citation Context ...(b ⌣1 c). Main examples of hga’s are: C ∗ (X) (see [2], [3],[13] and previous section) and the Hochschild cochain complex of an associative algebra where E1,1 and E2,1 were defined by Gerstenhaber in =-=[11]-=- and the higher operations were described in ([17], [13]). One more example is the cobar construction of a dg Hopf algebra [18]. Note also that certain algebras (including polynomial ones), which are ... |

138 |
A Basic Course in Algebraic Topology
- Massey
(Show Context)
Citation Context ..., satisfying the standard conditions [15]. An example of a cubical set is the singular cubical set Sing I Y = {Sing I n Y }n≥0 of a space Y, where Sing I n Y is the set of all continuous maps In −→ Y =-=[22]-=-. Analogously to a simplicial set for a cubical set Q and an R-module A its chain complex in coefficients A is defined which will be denoted by ( ¯ C✷ ∗ (Q; A), d). The normalized chain complex (C✷ ∗ ... |

123 |
homotopy algebra and iterated integrals for double loop spaces
- Getzler, Jones, et al.
(Show Context)
Citation Context ...− E2,1(da, b; c) − (−1) |a| E2,1(a, db; c) − (−1) |a|+|b| E2,1(a, b; dc) = (−1) |a|+|b| ab ⌣1 c − (−1) |a|+|b||c| (a ⌣1 c)b − (−1) |a|+|b| a(b ⌣1 c). Main examples of hga’s are: C ∗ (X) (see [2], [3],=-=[13]-=- and previous section) and the Hochschild cochain complex of an associative algebra where E1,1 and E2,1 were defined by Gerstenhaber in [11] and the higher operations were described in ([17], [13]). O... |

51 |
Homologie singulière des espaces fibrés
- Serre
- 1951
(Show Context)
Citation Context ...rod chain (co)operations and other (co)chain operations. In this paper we concentrate only on the strictly coassociative Serre diagonal (the cubical analog of the Alexander-Whitney (AW) diagonal, see =-=[30]-=-). The combinatorial analysis of the Serre diagonal allows us to give explicit formulas for a strictly associative multiplication on the twisted tensor product in terms of the ⌣1-product and other rel... |

49 |
E.H.: Twisted tensor products I
- Brown
- 1959
(Show Context)
Citation Context ...ory of twisted tensor products for homotopy G-algebras allows to obtain multiplicative models for fibrations. 1. Introduction For a fibration F → E → Y on the tensor product C∗ (Y ) ⊗ C∗ (F) E. Brown =-=[8]-=- has introduced a twisted differential dτ such that the homology of the cochain complex (C∗(Y ) ⊗ C∗ (F), dτ) is isomorphic to the cohomology H∗ (E) but just additively. So there arises the problem of... |

41 |
On the Cobar construction
- Adams
- 1956
(Show Context)
Citation Context ...f a 1-reduced simplicial set X.A CUBICAL MODEL FOR A FIBRATION 9 5. The cubical model of the path space fibration Let ΩY i −→ PY π −→ Y be the Moore path space fibration on a topological space Y. In =-=[1]-=- Adams constructed a dga map ω∗ : ΩC∗(Y ) −→ C ✷ ∗ (ΩY ) (1) being a weak equivalence for simply connected Y. It appears that Adams’ cobar construction and the above map ω∗ as well as the acyclic coba... |

41 |
On the chain complex of a fibration
- Gugenheim
- 1972
(Show Context)
Citation Context ...s in the Definition 4.2. We have that a truncating twisting function τ induces on chain level the twisting cochains τ∗ : C∗(X) −→ C∗−1(Q) and τ ∗ : C ∗ (Q) −→ C ∗+1 (X) in the standard sense ([8],[7],=-=[14]-=-). It is straightforward to verify that we have the equality and, consequently, the inclusion C ✷ ∗ (X ×τ L) = C∗(X) ⊗τ∗ C ✷ ∗ (L) (3) C ∗ ✷(X ×τ L) ⊃ C ∗ (X) ⊗τ ∗ C∗ ✷(L) (4) of dg modules (where we ... |

38 |
Iterated loop spaces
- Milgram
(Show Context)
Citation Context ... and the above map ω∗ as well as the acyclic cobar construction Ω(C∗(Y ); C∗(Y )) have explicit combinatorial interpretations by means of cubical sets. Namely, we have the following theorem (compare, =-=[23]-=-, [9], [2], [3] [10]). Theorem 5.1. (i) For the fibration ΩY i −→ PY π −→ Y there is the following commutative diagram Sing I ΩY ↑ ω⏐ i∗ −−−−→ Sing I PY p ↑ ⏐ Ω Sing 1 Y −−−−→ PSing 1 Y π∗ −−−−→ Sing ... |

35 |
Applications of perturbation theory to iterated fibrations
- Lambe, Stasheff
(Show Context)
Citation Context ...ebra too. In this way various multiplicative models were constructed in which either the associativity was abolished or a differential was not a derivation (see, for example, L. Lambe and J. Stasheff =-=[21]-=- for references). The standard notion of a twisting function τ : X∗ → G∗−1 from a simplicial set to a simplicial group and the notion of corresponding twisted Cartesian product X ×τ G does not allow t... |

30 |
Geometry of loop spaces and the cobar construction
- Baues
(Show Context)
Citation Context ... the comultiplication on C∗(X) ⊗τ∗ C✷ ∗ (L) by canonical higher order chain operations of degree k E k,1 : C∗(X) −→ C∗(X) ⊗k ⊗ C∗(X), k ≥ 0, on C∗(X), which agree with operations constructed by Baues =-=[2]-=-, [3] on the normalized complex C N ∗ (X), and by the action C✷ ∗ (Q) ⊗ C✷ ∗ (L) −→ C✷ ∗ (L) involving also τ∗. Interestingly, τ∗ takes place in both definition of a twisted differential and of a twis... |

22 |
Homologie singulière des espaces fibrés. Applications
- Serre
- 1951
(Show Context)
Citation Context ...ts of Q. Note also that both ¯C ✷ ∗ (Q) and C✷ ∗ (Q) are DG-coalgebras with respect to the canonical comultiplication determined by the Cartesian product decomposition of the n-cube In = I × · · · × I=-=[27]-=-. For a space Y we will denote C✷ ∗ (SingI Y ; Z) by C✷ ∗ (Y ). The (tensor) product of two cubical sets Q and Q ′ is Q × Q ′ = {(Q × Q ′ )n = ⋃ p+q=n Qp × Q ′ q }/ ∼ where (ηp+1(a), b) ∼ (a, η1(b)), ... |

14 |
On the differentials of spectral sequences
- Berikashvili
(Show Context)
Citation Context ...re as in the Definition 4.2. We have that a truncating twisting function τ induces on chain level the twisting cochains τ∗ : C∗(X) −→ C∗−1(Q) and τ ∗ : C ∗ (Q) −→ C ∗+1 (X) in the standard sense ([8],=-=[7]-=-,[14]). It is straightforward to verify that we have the equality and, consequently, the inclusion C ✷ ∗ (X ×τ L) = C∗(X) ⊗τ∗ C ✷ ∗ (L) (3) C ∗ ✷(X ×τ L) ⊃ C ∗ (X) ⊗τ ∗ C∗ ✷(L) (4) of dg modules (wher... |

12 |
Abstract homotopy
- Kan
- 1955
(Show Context)
Citation Context ...cubical set is a sequence of sets Q = {Qn}n≥0 with boundary operators dǫ i : Qn → Qn−1, ǫ = 0, 1, 1 ≤ i ≤ n, and degeneracy operators ηi : Qn → Qn+1, 1 ≤ i ≤ n + 1, satisfying the standard conditions =-=[15]-=-. An example of a cubical set is the singular cubical set Sing I Y = {Sing I n Y }n≥0 of a space Y, where Sing I n Y is the set of all continuous maps In −→ Y [22]. Analogously to a simplicial set for... |

11 |
Stable homotopy and iterated loop spaces', Handbook of algebraic topology
- Carlsson, Milgram
- 1995
(Show Context)
Citation Context ...equired multiplication. Finally, we mention that the geometric realization |ΩSing 1 Y | of ΩSing 1 Y is homeomorphic to the cellular model for the loop space observed by G. Carlsson and R. J. Milgram =-=[9]-=-. In [2], [3] H.-J. Baues has defined a geometric coassociative and homotopy cocommutative diagonal on the cobar construction ΩCN ∗ (Y ) by means of a certain cellular model for the loop space (homoto... |

7 |
Perturbation and obstruction theories in fibre spaces
- Saneblidze
(Show Context)
Citation Context ...rojection BA −→ A provides an example of the universal multiplicative element. For a commutative dga A one has the equality µE = µsh, so Proute’s twisting element is multiplicative (see, for example, =-=[25]-=-). We have that the argument of the proof of formula (6) immediately yields Theorem 7.1. Let τ ∗ : C −→ A be a multiplicative twisting element. Then the tensor product A⊗M with the canonical twisting ... |

4 | On the homology of the Whitney sum of fibre spaces - Khelaia - 1986 |

3 |
A∞-structures, Modele minimal de Bauess-Lemaire des fibrations, preprint
- Proute
(Show Context)
Citation Context ...a A is equivalent to a dg Hopf algebra structure on the (co)bar construction (ΩA)BA. We develop the theory of multiplicative twisted tensor products for homotopy G-algebras in the following sense. In =-=[24]-=- it is shown that if a twisting element φ : C → A from a dg Hopf algebra C to a commutative dga A is coprimitive, that is if the induced map C → BA is a map of dg Hopf algebras (where on BA the standa... |

3 | The homology of twisted cartesian products, Trans - Szczarba - 1961 |

2 |
DG Hopf algebras with Steenrod’s i-th coproducts
- Kadeishvili
(Show Context)
Citation Context ... we obtain the above multiplicative twisted tensor product [6]. Note also that in this multiplicative model it could be introduced Steenrod’s (co)chain operations as they are defined for cubical sets =-=[16]-=-. By using the standard triangulation of the cubes one obtains a map of dg Hopf algebras C∗ N (G) → C∗ ✷(G) and then it is possible to obtain again multiplicative twisted tensor product C∗ (Y ) ⊗τ ′∗ ... |

2 | The twisted Cartesian model for the double path space fibration, preprint
- Kadeishvili, Saneblidze
- 2002
(Show Context)
Citation Context ...e fiber respectively. In this paper we begin to develop the general theory of twisting functions to form twisted Cartesian products of abstract sets of different kind. The continuation will follow in =-=[19]-=- where twisted functions from cubical sets to permutahedral sets will be considered. Here we introduce the notion of a truncating twisting function τ : X∗ → Q∗−1 where X is a 1-reduced simplicial set ... |

1 | An algebraic model of fibration with the fiber K(π, n)-space
- Berikashvili
- 1996
(Show Context)
Citation Context ...ituation radically changes if we replace simplicial group G by a monoidal cubical set and suitably modify the notion of a twisting function. This idea comes from recent results of N. Berikashvili: In =-=[5]-=- a multiplicative model with associative multiplication in the case when the fiber F is the cubical version of the Eilenberg-MacLane space is constructed; in the next paper [6] a multiplicative model ... |

1 |
The multiplicative version of twisted tensor product theorem
- Berikashvili, Makalatia
- 1996
(Show Context)
Citation Context ... of N. Berikashvili: In [5] a multiplicative model with associative multiplication in the case when the fiber F is the cubical version of the Eilenberg-MacLane space is constructed; in the next paper =-=[6]-=- a multiplicative model C∗ (Y ) ⊗φ C∗ ✷ (F), φ : C∗ ✷ (G) → C∗+1 (Y ) is constructed where C∗ (Y ) is the singular simplicial cochain complex of the base and C∗ ✷ (G) and C∗ ✷ (F) are the singular cub... |

1 |
The homotopy type FΨ q , the complex and sympletic cases, Cont
- Huebschmann
- 1986
(Show Context)
Citation Context ... Award No. 99-00817 of INTAS. 12 TORNIKE KADEISHVILI AND SAMSON SANEBLIDZE twisted tensor product C∗(X) ⊗φ C∗(N). But the resulting (co)multiplication is (co)associative only up to higher homotopies =-=[15]-=-,[23]. The situation changes radically if we replace a simplicial group G by a monoidal cubical set and suitably modify the notion of a twisting function. This yields a cubical model of a fibration wh... |

1 |
Fibre bundles, third ed
- Husemoller
- 1994
(Show Context)
Citation Context ...ltiplication which extends the one from H∗(X) multiplicatively, and the Bott-Samelson map i∗ is a Hopf algebra isomorphism too. There is the dual statement for the cohomology as well (cf. Appendix in =-=[16]-=-). First we recover the above facts in the following way. Let Y be the suspension over a polyhedron X; explicitly, regard Y as the geometric realization of a quotient simplicial set Y = SX/C−X where S... |

1 |
operations defining Steenrod ⌣i-products in the bar construction
- Cochain
(Show Context)
Citation Context ...) ⊗τ ∗ C∗ □ (L) ⊂ C∗ □ (X ×τ L) (here we have equality when the graded sets have finite type). Also note that the chain operations dual to Steenrod ⌣i operations are defined for cubical sets in [18], =-=[19]-=- and the equality C□ ∗ (ΩX) = ΩC∗(X) allows to define these operations on the cobar construction ΩC∗(X); similarly since C□ ∗ (X ×τ L) = C∗(X) ⊗τ∗ C□ ∗ (L) it is possible to introduce Steenrod operati... |