On Grothendieck's tame topology

Norbert A'Campo (1, 2) , Lizhen Ji (3, 2) , Athanase Papadopoulos (4, 2)

1 Unibas - Université de Bâle = University of Basel = Basel Universität
2 Institut Mittag-Leffler
3 Department of Mathematics - University of Michigan
4 IRMA - Institut de Recherche Mathématique Avancée

Alexander Grothendieck, motivated by surface topology and moduli spaces of Riemann surfaces, calls in his in his ``Esquisse d'un programme" for a recasting of topology, in order to make it fit to the objects of semialgebraic and semianalytic geometry, and in particular to the study of the Mumford-Deligne compactifications of moduli spaces. A new conception of manifold, of submanifold and of maps between them is outlined. We review these ideas in the present chapter, because of their relation to the theory of moduli and Teichmüller spaces. We also mention briefly the relations between Grothendieck's ideas and earlier theories developed by Whitney, Lojasiewicz and Hironaka and especially Thom, and with the more recent theory of o-minimal structures.

Mots clés: Modular multiplicity; tame topology; stratified structure; moderate topology; o-minimal structure; semialgebraic geometry; semianalytic geometry; subanalytic geometry

Domaines: Topologie géométrique [math.GT]; Géométrie algébrique [math.AG]

Norbert A'Campo, Lizhen Ji, Athanase Papadopoulos. On Grothendieck's tame topology. Handbook of Teichmüller theory, Volume VI, 2016, Handbook of Teichmüller theory, Vol. VI, European Mathematical Society, Zurich, 2016, pp. 521--532., 978-3-03719-161-3. ⟨10.4171/161⟩. ⟨hal-01346539⟩