2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/155874We consider the quantum Teichmuller space of the punctured surface introduced by Chekhov-Fock-Kashaev, and formalize it as a noncommutative deformation of the space of algebraic functions on the Teichmuller space of the surface. In order to apply it in 3-dimensional topology, we put more attention to the details involving small surfaces.Revised version. 15 pages, 4 figures, AmsLaTeX file. A few misprints correctedGeometric TopologyQuantum Algebra57R56 (Primary); 57M50, 20G42 (Secondary)text