## Rocío González–Díaz, Pedro Real Universidad de Sevilla, Depto. de Matemática Aplicada I, (2001)

### BibTeX

@MISC{01rocíogonzález–díaz,,

author = {},

title = {Rocío González–Díaz, Pedro Real Universidad de Sevilla, Depto. de Matemática Aplicada I,},

year = {2001}

}

### OpenURL

### Abstract

In this note, working in the context of simplicial sets [17], we give a detailed study of the complexity for computing chain level Steenrod squares [20, 21], in terms of the number of face operators required. This analysis is based on the combinatorial formulation given in [5]. As an application, we give here an algorithm for computing cup–i products over integers on a simplicial complex at chain level. 1

### Citations

454 |
Elements of Algebraic Topology
- Munkres
- 1984
(Show Context)
Citation Context ...where m = i + j and n = j − i. ⎛ ⎜ ⎝ ⌊ ⌋ m 2 ⌊ ⌋ n 2 ⎞ ⎛ ⎟ ⎜ ⎠ ⎝ ⌊ ⌋ m+1 2 ⌊ ⌋ n+1 2 ⎞ ⎟ ⎠ , 4 Simplicial Complexes Now, let us study a particular simplicial set. A (combinatorial) simplicial complex =-=[18, 22]-=- is a collection P of nonempty finite subsets of some vertex set V such that if τ ⊂ σ ⊂ V and σ ∈ P, then τ ∈ P. If the vertex set is ordered, we call P an ordered simplicial complex. To every such or... |

405 |
An introduction to homological algebra, Cambridge Studies in Advanced Mathematics 38, Cambridge
- Weibel
- 1994
(Show Context)
Citation Context ...where m = i + j and n = j − i. ⎛ ⎜ ⎝ ⌊ ⌋ m 2 ⌊ ⌋ n 2 ⎞ ⎛ ⎟ ⎜ ⎠ ⎝ ⌊ ⌋ m+1 2 ⌊ ⌋ n+1 2 ⎞ ⎟ ⎠ , 4 Simplicial Complexes Now, let us study a particular simplicial set. A (combinatorial) simplicial complex =-=[18, 22]-=- is a collection P of nonempty finite subsets of some vertex set V such that if τ ⊂ σ ⊂ V and σ ∈ P, then τ ∈ P. If the vertex set is ordered, we call P an ordered simplicial complex. To every such or... |

265 |
Simplicial objects in algebraic topology
- May
- 1967
(Show Context)
Citation Context ...Universidad de Sevilla, Depto. de Matemática Aplicada I, Avda. Reina Mercedes, 41012 Sevilla, Spain, e-mails: rogodi@us.es, real@us.es Abstract In this note, working in the context of simplicial sets =-=[17]-=-, we give a detailed study of the complexity for computing chain level Steenrod squares [20, 21], in terms of the number of face operators required. This analysis is based on the combinatorial formula... |

78 |
Algebraic topology, McGraw-Hill
- Spanier
- 1966
(Show Context)
Citation Context ...computing cup–i products over integers on a simplicial complex at chain level. 1 Introduction Cohomology operations are tools for calculating n-cocycles on the cohomology of spaces (see, for example, =-=[16, 19]-=-). Unfortunately, up to the present, no symbolic computational system includes general methods for finding representative n–cocycles on the cohomology of spaces, algebras, groups, etc. Recently, sever... |

42 |
A history of Algebraic and Differential Topology
- Dieudonné
- 1989
(Show Context)
Citation Context ...gree (see [1] and [6]). A treatment of some of our methods has already been presented in [7]. In the literature, there is plenty of information about cup–i products and Steenrod squares (see [23] and =-=[3]-=- for a non–exhaustive account of results). We think that the algorithmic technique explained here could be substantially refined if it is suitably combined with relevant and well–known results on thes... |

41 |
On the chain-complex of a fibration
- Gugenheim
(Show Context)
Citation Context ...ned if it is suitably combined with relevant and well–known results on these cohomology operations and with techniques of homological perturbation for manipulating explicit homotopy equivalences (see =-=[2, 8, 9, 10]-=-). We are grateful to Prof. Julio Rubio for his helpful suggestions for improving the algorithms showed here. 2 Topological and Algebraic Preliminaries The aim of this section is to give some simplici... |

32 |
The twisted Eilenberg–Zilber theorem. Celebrazioni Archimedee del secolo XX, Simposio di Topologia
- Brown
- 1967
(Show Context)
Citation Context ...ned if it is suitably combined with relevant and well–known results on these cohomology operations and with techniques of homological perturbation for manipulating explicit homotopy equivalences (see =-=[2, 8, 9, 10]-=-). We are grateful to Prof. Julio Rubio for his helpful suggestions for improving the algorithms showed here. 2 Topological and Algebraic Preliminaries The aim of this section is to give some simplici... |

24 |
Products of cocycles and extensions of mappings
- Steenrod
- 1947
(Show Context)
Citation Context ..., Spain, e-mails: rogodi@us.es, real@us.es Abstract In this note, working in the context of simplicial sets [17], we give a detailed study of the complexity for computing chain level Steenrod squares =-=[20, 21]-=-, in terms of the number of face operators required. This analysis is based on the combinatorial formulation given in [5]. As an application, we give here an algorithm for computing cup–i products ove... |

21 | Problems in the Steenrod algebra
- Wood
- 1998
(Show Context)
Citation Context ...in any degree (see [1] and [6]). A treatment of some of our methods has already been presented in [7]. In the literature, there is plenty of information about cup–i products and Steenrod squares (see =-=[23]-=- and [3] for a non–exhaustive account of results). We think that the algorithmic technique explained here could be substantially refined if it is suitably combined with relevant and well–known results... |

19 |
Singular homology theory, in: Graduate Texts
- Massey
- 1980
(Show Context)
Citation Context ...computing cup–i products over integers on a simplicial complex at chain level. 1 Introduction Cohomology operations are tools for calculating n-cocycles on the cohomology of spaces (see, for example, =-=[16, 19]-=-). Unfortunately, up to the present, no symbolic computational system includes general methods for finding representative n–cocycles on the cohomology of spaces, algebras, groups, etc. Recently, sever... |

18 | Homological perturbation theory, Hochschild homology, and formal groups. Deformation theory and quantum groups with applications to mathematical physics
- Lambe
- 1990
(Show Context)
Citation Context ... for finding 2–cocycles representing 2–dimensional cohomology classes of finite groups have been designed (see [4, 12, 14]). The method established in [14] is based on the general theory presented in =-=[13]-=- and it seems that can be generalized to higher dimension without effort. In this paper, we describe a different procedure based on a combinatorial formulation given in [5] for an important class of c... |

15 | Real P.: A combinatorial method for computing Steenrod squares
- Gonz'alez-D'iaz
- 1999
(Show Context)
Citation Context ...tailed study of the complexity for computing chain level Steenrod squares [20, 21], in terms of the number of face operators required. This analysis is based on the combinatorial formulation given in =-=[5]-=-. As an application, we give here an algorithm for computing cup–i products over integers on a simplicial complex at chain level. 1 Introduction Cohomology operations are tools for calculating n-cocyc... |

12 |
Reduced powers of cohomology classes
- Steenrod
- 1952
(Show Context)
Citation Context ..., Spain, e-mails: rogodi@us.es, real@us.es Abstract In this note, working in the context of simplicial sets [17], we give a detailed study of the complexity for computing chain level Steenrod squares =-=[20, 21]-=-, in terms of the number of face operators required. This analysis is based on the combinatorial formulation given in [5]. As an application, we give here an algorithm for computing cup–i products ove... |

11 |
Cocyclic development of designs
- Horadam, Launey
- 1993
(Show Context)
Citation Context ... n–cocycles on the cohomology of spaces, algebras, groups, etc. Recently, several methods for finding 2–cocycles representing 2–dimensional cohomology classes of finite groups have been designed (see =-=[4, 12, 14]-=-). The method established in [14] is based on the general theory presented in [13] and it seems that can be generalized to higher dimension without effort. In this paper, we describe a different proce... |

7 |
Algorithms for algebraic computations with applications to the cohomology of finite p-groups
- Grabmeier, Lambe
- 1997
(Show Context)
Citation Context ... n–cocycles on the cohomology of spaces, algebras, groups, etc. Recently, several methods for finding 2–cocycles representing 2–dimensional cohomology classes of finite groups have been designed (see =-=[4, 12, 14]-=-). The method established in [14] is based on the general theory presented in [13] and it seems that can be generalized to higher dimension without effort. In this paper, we describe a different proce... |

6 |
Perturbation and transfer of generic algebraic structure, Higher homotopy structures in topology and mathematical physics
- Hess
- 1996
(Show Context)
Citation Context ... Steenrod [20] for the cup–i product on simplicial complexes. We note that a mod–2 explicit formulation of the Steenrod coproduct on the chain of a simplicial set has also been given in (6.2) of Hess =-=[11]-=-, using a different method. ∗ Both authors are partially supported by the PAICYT research project FQM-0143 from Junta de Andalucía and the DGES-SEUID research project PB97-1025-C02-02 from Education a... |

5 |
On the Theory and
- Gugenheim, May
- 1974
(Show Context)
Citation Context ...ned if it is suitably combined with relevant and well–known results on these cohomology operations and with techniques of homological perturbation for manipulating explicit homotopy equivalences (see =-=[2, 8, 9, 10]-=-). We are grateful to Prof. Julio Rubio for his helpful suggestions for improving the algorithms showed here. 2 Topological and Algebraic Preliminaries The aim of this section is to give some simplici... |

3 |
P.: Algorithms in Algebraic Topology and Homological Algebra. The Problem of the Complexity. Extended Abstracts of the
- Álvarez, Armario, et al.
(Show Context)
Citation Context ...al complexes. We integrate here tools of Combinatorics and Computer Algebra in a work of Algebraic Topology, opening a door to a computational development in the search of cocycles in any degree (see =-=[1]-=- and [6]). A treatment of some of our methods has already been presented in [7]. In the literature, there is plenty of information about cup–i products and Steenrod squares (see [23] and [3] for a non... |

2 | P.: Una curiosa combinación de Topología, Álgebra y Combinatoria: los cuadrados de Steenrod. La Gaceta de la Real Sociedad Matemática Española 3 - González–Díaz, Real - 1999 |

2 |
P.: Computing the action of the Steenrod algebra on the cohomology of polyhedral simplicial sets
- González–Díaz, Real
- 1998
(Show Context)
Citation Context ...a work of Algebraic Topology, opening a door to a computational development in the search of cocycles in any degree (see [1] and [6]). A treatment of some of our methods has already been presented in =-=[7]-=-. In the literature, there is plenty of information about cup–i products and Steenrod squares (see [23] and [3] for a non–exhaustive account of results). We think that the algorithmic technique explai... |