Francis Sergeraert

Computing All Maps into a Sphere

Given topological spaces X,Y, a fundamental problem of algebraic topology is understanding the structure of all continuous maps X → Y. We consider a computational version, where X,Y are given as finite simplicial complexes, and the goal is to compute [X,Y], that is, all homotopy classes of such maps. We solve this problem in the stable range, where for some d ≥ 2, we have dim X ≤ 2d−2 and Y is (d-1)-connected; in particular, Y can be the d-dimensional sphere Sd. The algorithm combines classical tools and ideas from homotopy theory (obstruction theory, Postnikov systems, and simplicial sets) with algorithmic tools from effective algebraic topology (locally effective simplicial sets and objects with effective homology). In contrast, [X,Y] is known to be uncomputable for general X,Y, since for X=S1 it includes a well known undecidable problem: testing triviality of the fundamental group of Y. In follow-up papers, the algorithm is shown to run in polynomial time for d fixed, and extended to other problems, such as the extension problem, where we are given a subspace A ⊂ X and a map A → Y and ask whether it extends to a map X → Y, or computing the ℤ2-index—everything in the stable range. Outside the stable range, the extension problem is undecidable.

Computing spectral sequences

John McCleary insisted in his interesting textbook entitled “User’s guide to spectral sequences” on the fact that the tool “spectral sequence” is not in the general situation an algorithm allowing its user to compute the looked-for homology groups. The present article explains how the notion of “Object with Effective Homology” on the contrary allows the user to recursively obtain all the components of the Serre and Eilenberg‐Moore spectral sequences, when the data are objects with effective homology. In particular the computability problem of the higher differentials is solved, the extension problem at abutment is also recursively solved. Furthermore, these methods have been concretely implemented as an extension of the Kenzo computer program. Two typical examples of spectral sequence computations are reported. c 2006 Elsevier Ltd. All rights reserved.

fKenzo: A user interface for computations in Algebraic Topology

a b s t r a c t fKenzo (= f riendly Kenzo) is a graphical user interface providing a user-friendly front-end for the Kenzo system, a Common Lisp pro- gram devoted to Algebraic Topology. The fKenzo system provides the user interface itself, an XML intermediary generator-translator and, finally the Kenzo kernel. We describe in this paper the main points of fKenzo, and we explain also the advantages and limita- tions of fKenzo with respect to Kenzo itself. The text is separated into two parts, trying to cover both the user and the developer perspectives.

Locally effective objects and algebraic topology

Algebraic topology consists of associating invariants are of an algebraic nature, describing certain topological properties. For example, since Poincare, it is known how to associate the group π1(X,x 0 to a topological space X and to one of its points x 0; this group is called the Poincare group or the first homotopy group of the space X based on x 0. This group is null if and only if the space X is simply connected at x 0; in another case, the group measures the lack of simple connectivity. Many other groups can be associated to a topological space, evaluating certain properties of this space: homology groups,K-theory groups,etc.

Constructive algebraic topology

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.  2002 Editions scientifiques et medicales Elsevier SAS. All rights reserved.

Effective homotopy of fibrations

The well-known effective homology method provides algorithms computing homology groups of spaces. The main idea consists in keeping systematically a deep and subtle connection between the homology of any object and the object itself. Now applying similar ideas to the computation of homotopy groups, we aim to develop a new effective homotopy theory which allows one to determine homotopy groups of spaces. In this work we introduce the notion of a solution for the homotopical problem of a simplicial set, which will be the main definition of our theory, and present an algorithm computing the effective homotopy of a fibration. We also illustrate with examples some applications of our results.

A program computing the homology groups of loops spaces

The object of this note is to describe the second version of a program allowing in theory the possibility of calculating the homology groups of iterated loop spaces <i>H</i><inf><i>q</i></inf> (&Omega;^<i>p</i> X; <b>Z</b>) (solving every extension problem) if <i>X</i> is a reduced simplicial set with effective homology the skeleton of which begins in dimension <i>p</i>+1. It is understood, bearing in mind the hyper-exponential complexity of the algorithm, an overambitious choice for <i>q, p</i> and <i>X</i> would not allow the obtaining of a result in a reasonable time. But several examples of calculations already made with the help of this program show that it is possible to find groups, which up until now, have seemed to be inaccessible.

A Combinatorial Tool for Computing the Effective Homotopy of Iterated Loop Spaces

This paper is devoted to the Cradle Theorem. It is a recursive description of a discrete vector field on the direct product of simplices

A case study of ∞-structure

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