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# Ecole Été 1997

Fundamental Algebraic Topology

The following lecture notes are available : Clemens Berger : Espaces de lacets itérés et groupes symétriques. PostScript file. Ronnie Brown : Groupoids and crossed objects in algebraic topology. PostScript file. Yves Félix : Sullivan minimal models. Dvi file. Takuji Kashiwabara : Homology of infinite loop spaces and cohomology operations. Dvi and PostScript files. Jean-Louis Loday : La renaissance des opérades. Dvi and PostScript files. Lionel Schwartz : Unstable modules and unstable algebras ; applications to homotopy theory. Dvi and PostScript files. Francis Sergeraert : Topological constructors. Dvi and PostScript files. Effective homology, a survey. Dvi and PostScript files. Constructive algebraic topology. Dvi and PostScript files. Hans Baues : Homotopy types. Not available here : the lecture notes of Hans Baues’ courses was the paper Homotopy Types, pages 1-72 of James’ Handbook of Algebraic Topology, North-Holland, 1995.

Contents of these papers

Clemens Berger : Espaces de lacets itérés et groupes symétriques. PostScript file. Reconnaître et reconstruire un espace de lacets itérés. Structure cellulaire des E_n-opérades. Homologie du groupe symétrique infini.

Ronnie Brown : Groupoids and crossed objects in algebraic topology. PostScript file. Introduction. Groupoids in homotopy theory. The search for higher order versions of the fundamental group in homotopy theory. Whitehead’s work on crossed modules. The 2-dimensional Van Kampen theorem. Computation with crossed modules and second relative homotopy groups. Preliminaries on double groupoids. The homotopy double groupoid of a triple of spaces. The 2-dimensional Van Kampen theorem. The category of crossed complexes. Homotopies of morphisms of crossed complexes. Relations with chain complexes with operators. Cohomology classes. Further work.

Yves Félix : Sullivan minimal models. Dvi file. The first homotopy groups of a 1-connected space. The dichotomy theorem. Fibrations. Adams-Hilton models. Spaces of category 2.

Takuji Kashiwabara : Homology of infinite loop spaces and cohomology operations. Dvi and PostScript files. Stable and unstable cohomology operations. Hoph rings and homology of infinite loop spaces. Algebras over unstable operations. For further study.

Jean-Louis Loday : La renaissance des opérades. Dvi and PostScript files. Introduction. Opérades K-linéaires. Dualité de Koszul dans les algèbres associatives. Dual d’une opérade quadratique. Dualité de Koszul des opérades. Applications.

Lionel Schwartz : Unstable modules and unstable algebras ; applications to homotopy theory. Dvi and PostScript files. The mod 2 Steenrod algebra \cal A_2 ; unstable modules and unstable algebras. Injective objects in \cal U ; Lannes’ T functor. Applications of the functor T.

Francis Sergeraert : Topological constructors. Dvi and PostScript files. Introduction. CW-complexes. The category \Delta. Simplicial sets.

Francis Sergeraert : Effective homology, a survey. Dvi and PostScript files. Introduction. Is a spectral sequence an algorithm ? Functional algorithmic, a survey. Objects with effective homology. Perturbation lemma machinery. Rubio’s solution to Adam’s problem. A program.

Francis Sergeraert Constructive algebraic topology. Dvi and PostScript files. Introduction. General organization. Functional programming in Lisp. Functional programming and infinite simplicial sets. A mathematical definition is not necessarily a constructive definition. The EAT program. EAT and the loop spaces. A tentative appropriate mathematical language. The general organization revisited. The basic perturbation lemma. Bicomplexes and cones. Tensor products. Fibrations. The twisted Eilenberg-Zilber theorem. The SERRE_EH algorithm.

Web Page of F. Sergeraert