## Constructive algebraic topology (1999)

Venue: | SIGSAM Bull |

Citations: | 8 - 5 self |

### BibTeX

@ARTICLE{Rubio99constructivealgebraic,

author = {Julio Rubio and Francis Sergeraert},

title = {Constructive algebraic topology},

journal = {SIGSAM Bull},

year = {1999},

pages = {13--25}

}

### OpenURL

### Abstract

The classical “computation ” methods in Algebraic Topology most often work by means of highly infinite objects and in fact are not constructive. Typical examples are shown to describe the nature of the problem. The Rubio-Sergeraert solution for Constructive Algebraic Topology is recalled. This is not only a theoretical solution: the concrete computer program Kenzo has been written down which precisely follows this method. This program has been used in various cases, opening new research subjects and producing in several cases significant results unreachable by hand. In particular the Kenzo program can compute the first homotopy groups of a simply connected arbitrary simplicial set.

### Citations

123 |
A User’s Guide to Spectral Sequences
- McCleary
- 2001
(Show Context)
Citation Context ...severe. The three current solutions [17, 19, 11, 23, 24] for Constructive Algebraic Topology are firstly solutions for the problem of iterating the cobar construction. John McCleary tries in his book =-=[15]-=- to express the same idea in the context of spectral sequences: [p. 6] “Theorem”. There is a spectral sequence with E ∗,∗ 2 = “something computable” and converging to H∗, something desirable. The impo... |

94 |
The homology of iterated loop spaces
- Cohen, Lada, et al.
- 1976
(Show Context)
Citation Context ...X is a suspension X = S n Y ; then the homology groups H∗Ω n X are entirely determined by the homology groups H∗Y thanks to a process where the Dyer-Lashof homology operations play the main role, see =-=[6, 7]-=-. For example the Moore space Moore(Z2,4) is nothing but the third suspension S 3 P 2 R, so that the homology groups H∗Ω 3 Moore(Z2,4) are entirely determined by the well known groups H∗P 2 R = (Z, Z2... |

40 |
On the cobar construction
- Adams
- 1956
(Show Context)
Citation Context ...the canonical simplicial structure of X 2 ; on the contrary the other chain complex C∗(X)⊗C∗(X) comes from the canonical bisimplicial structure of the same space. If for example X is the interval I = =-=[0,1]-=-, in the first case a square is presented as the union of two triangles joined along a diagonal; in the second case no diagonal in the square, only the boundary edges, the square is simply the product... |

29 |
Geometry of loop spaces and the cobar construction
- Baues
- 1980
(Show Context)
Citation Context ...ellular chain complex of a CW model of the iterate loop space Ω n X. The existence of the differential δ can be easily proved thanks to Adams’ work about the CW model of the first loop space (cf also =-=[2]-=-), but the existence proof is not constructive: it is made of a mixture of combinatorial and topological arguments and certainly there are at least “severe difficulties” to translate the topological c... |

20 | The computability problem in algebraic topology
- Sergeraert
- 1994
(Show Context)
Citation Context ...e cobar construction is the heart of Algebraic Topology: the main computability problems can be reduced to this one, and it is not amazing this problem is a little severe. The three current solutions =-=[17, 19, 11, 23, 24]-=- for Constructive Algebraic Topology are firstly solutions for the problem of iterating the cobar construction. John McCleary tries in his book [15] to express the same idea in the context of spectral... |

11 |
Finite computability of Postnikov complexes
- Brown
- 1957
(Show Context)
Citation Context ... exist a general algorithm: • Input: A simply connected polyhedron X and an integer n ≥ 2; • Output: The homotopy group πnX. A solution for this computability problem was given by Edgar Brown in 1956 =-=[3]-=-. He used the general organization just defined by Postnikov, now known as the Postnikov tower; then the result is not difficult when the homology groups of the space X are finite; really finite, not ... |

9 |
Stable homotopy and iterated loop spaces
- Carlsson, Milgram
- 1995
(Show Context)
Citation Context ...onstruction is the heart of Algebraic Topology and that many problems can be reduced to problems about loop spaces; they were invented by Jean-Pierre Serre fifty years ago for this reason. 8 See also =-=[6]-=- for an excellent recent extensive study of the subject. 199 Examples of calculations. 9.1 H5Ω 3 Moore(Z2, 4). Carlsson and Milgram explain in the paper quoted in Section 1 the computation of H∗Ω n X... |

9 |
Iterating the cobar construction
- Smith
(Show Context)
Citation Context ...roubi is not yet reached by his method. In fact three solutions now exist for Constructive Algebraic Topology, and two of them exactly have the form that Karoubi looked for. In Justin Smith’ solution =-=[23, 24]-=-, the cellular chain complex Ccell ∗ (X) is provided with a m-structure which, in appropriate context, is a computable algebraic model for the homotopy type of X. In the Rubio-S. solution, the same ch... |

7 | The twisted Eilenberg-Zilber theorem
- Brown
- 1964
(Show Context)
Citation Context ...lenberg-Moore spectral sequences are available in the program Kenzo. The main ingredient for the proof of the effective homology version of the Serre spectral sequence is the Basic Perturbation Lemma =-=[5]-=-. Theorem 5.2. Basic Perturbation Lemma — Let ρ : D∗ ⇒ C∗ be a chain complex reduction and δD∗ : D∗ → D∗ a perturbation of the differential dD∗ satisfying the nilpotency condition. Then a general algo... |

6 |
Effective algebraic topology
- Schön
- 1991
(Show Context)
Citation Context ...del for the homotopy type of X. In the Rubio-S. solution, the same chain complex is completed with two other chain complexes and a few operators which give the same result. The solution by Rolf Schön =-=[17]-=- is not presented in this way but finally is equivalent to both previous ones. Let us quote now a paper by Carlsson and Milgram in James’ Handbook of Algebraic Topology [6, p. 545]: In Section 5 we sh... |

6 | Collected papers - Serre - 2000 |

5 |
The kenzo program. http://www-fourier.ujf-grenoble.fr/ ~sergerar/Kenzo
- Dousson, Rubio, et al.
- 2008
(Show Context)
Citation Context ...n Edgar Brown’s one. Section 7 describes how these theoretical results led to a concrete program named Kenzo 1 . It is a Lisp program of 16000 lines (joint work with Xavier Dousson), now wwwavailable =-=[11]-=- with a rich documentation (340 pp) written by Yvon Siret. These results open new research fields; in Computer Science because of the original type of functional programming which is required, but in ... |

4 |
Homologie des espaces fibrés. Publications Mathématiques de l’Institut des Hautes Etudes Scientifiques
- Shih
- 1962
(Show Context)
Citation Context ...C∗(T) = C∗(B ×τ F) ⇒ EB∗ ⊗t EF∗ which describes the homology of the total space of the fibration by means of a twisted tensor product of the chain complexes EB∗ and EF∗. This was already done by Shih =-=[22]-=- and the present work about effective homology is nothing but the following remark: if functional programming is used, then Shih’s presentation of the Serre spectral sequence becomes an algorithm comp... |

2 |
Twisted tensor products
- Jr
(Show Context)
Citation Context ... twisted tensor product X ⊗t C∗ΩX is acyclic and an explicit contraction h of this chain complex plays a capital role in effective homology. The existence of this contraction is known for a long time =-=[4]-=-, but the explicit Kenzo computation of h shows very surprising properties, which imply we are far from mastering the underlying algebraic structure. Let us recall the loop space construction is the h... |

2 |
Sur la Cobar construction
- Dancète
- 1998
(Show Context)
Citation Context ...e result has not yet been proved. In particular it was completely obtained if a new differential is installed on Cobar C∗X (Z, Z), but it is not clear what the status of this new differential is. See =-=[9, 10]-=-. Other amazing experimental results of this sort have been obtained, in particular around the canonical algebraic fibration: C∗ΩX → X ⊗t C∗ΩX → X. This is the algebraic version of the co-universal fi... |

2 | m-structures determine integral homotopy type. http://vorpal.mcs.drexel.edu/research/m-homotop.pdf
- Smith
(Show Context)
Citation Context ...roubi is not yet reached by his method. In fact three solutions now exist for Constructive Algebraic Topology, and two of them exactly have the form that Karoubi looked for. In Justin Smith’ solution =-=[23, 24]-=-, the cellular chain complex Ccell ∗ (X) is provided with a m-structure which, in appropriate context, is a computable algebraic model for the homotopy type of X. In the Rubio-S. solution, the same ch... |

1 |
Sur l’itération de la construction Cobar. Comptes-Rendus de l’Académie des Sciences de
- Dancète
- 1999
(Show Context)
Citation Context ...e result has not yet been proved. In particular it was completely obtained if a new differential is installed on Cobar C∗X (Z, Z), but it is not clear what the status of this new differential is. See =-=[9, 10]-=-. Other amazing experimental results of this sort have been obtained, in particular around the canonical algebraic fibration: C∗ΩX → X ⊗t C∗ΩX → X. This is the algebraic version of the co-universal fi... |

1 |
Algèbres et cogèbres graduées avec symétries
- Karoubi
- 1993
(Show Context)
Citation Context ...be studied by hand, specially around algebraic fibrations. These questions are considered in Section 8. Finally Section 9 gives a few examples of calculations. 1 Three examples. A preprint by Karoubi =-=[14]-=-, distributed in 1993, begins as follows: The problem of finding a “computable algebraic model” for the homotopy type of a CW-complex X remains a widely open problem in topology. The notion of computa... |

1 |
The EAT program. ftp://www-fourier.ujf-grenoble.fr/~ftp/EAT
- Rubio, Sergeraert, et al.
(Show Context)
Citation Context ... �� ❅ ❅❅ ❅ ❅❅ � �� � u �� �v � w �� ❅ ❅❅ ❅ ❅❅ G G1 u �� � vw ��❅❅ ❅ 2 The first announcement goes back to 1987 [18]; the first computer program computing an iterate cobar construction started in 1990 =-=[16]-=-. � G2 10Let us consider the chromatic polynomial PG(X); it is a polynomial with one variable defined as follows: if n is a positive integer, then PG(n) is the number of good colourings of G that are... |

1 |
Homologie effective. Comptes-Rendus hebdomadaires des séances de l’Académie des Sciences
- Sergeraert
(Show Context)
Citation Context ...: the chromatic number does not contain enough information; we need more. � u �� �v α � w �� ❅ ❅❅ ❅ ❅❅ � �� � u �� �v � w �� ❅ ❅❅ ❅ ❅❅ G G1 u �� � vw ��❅❅ ❅ 2 The first announcement goes back to 1987 =-=[18]-=-; the first computer program computing an iterate cobar construction started in 1990 [16]. � G2 10Let us consider the chromatic polynomial PG(X); it is a polynomial with one variable defined as follo... |

1 |
k, objet du 3 e type
- Sergeraert
(Show Context)
Citation Context ...homotopy groups are ∗ This text was used as a background paper for a plenary talk of the second author during the EACA Congress of Tenerife, September 1999. A “general public” version has appeared in =-=[20]-=-; it is an excellent introduction for the present text. 1also of finite type. Various methods allow to combinatorially describe the finite polyhedra: these objects may be the input of an algorithm. A... |