On Grothendieck's tame topology

TL;DR

This paper explores Grothendieck's ideas on recasting topology to better understand moduli spaces, Teichmüller theory, and their connections with various geometric and analytic frameworks, highlighting a new conception of manifolds.

Contribution

It reviews Grothendieck's proposals for a new topology suited to semialgebraic and semianalytic geometry, connecting classical and modern theories in the context of moduli spaces.

Findings

Relation to Whitney, Lojasiewicz, Hironaka, Thom, and o-minimal structures

Proposal of a new conception of manifolds and maps

Insights into the study of moduli and Teichmüller spaces

Abstract

Grothendieck's Esquisse d'un programme is often referred to for the ideas it contains on dessins d'enfants, the Teichm{\"u}ller tower, and the actions of the absolute Galois group on these objects or their etale fundamental groups. But this program contains several other important ideas. In particular, motivated by surface topology and moduli spaces of Riemann surfaces, Grothendieck calls there 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 Te-ichm{\"u}ller 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. The final version of this paper will appear as a chapter in Volume VI of the Handbook of Teichm{\"u}ller theory. This volume is dedicated to the memory of Alexander Grothendieck.