Colloquium: "Classification of 2-manifolds" by Dr. Guilherme Sadovski

Speaker: Dr. Sadovski is a member of this unit (visit his page).

Abstract. In this talk, we review the concepts of a d-manifold in the topological (TOP), piece-wise linear (PL), differentiable (DIFF), analytical (C^ω) and biholomorphical (COMPLEX) categories. We start the study of 2-manifolds by stating that all obstruction invariants that prevent TOP 2-manifolds to be promoted to PL or DIFF or C^ω or COMPLEX ones are vanishing. We then proceed with the concept of Riemann surfaces and the very important result known as the Uniformization Theorem (UT). The UT allow us to state that all 2-manifolds are "geometrizable", i.e., admits a geometry and that these geometries can only be of three different (non-isometric) kinds. Using Gauss-Bonnet theorem, this fact is converted to "all closed TOP 2-manifolds can be completely characterized by the value of Euler invariant or, equivalently, by their number of handles (genus) and cross-caps.". We also briefly comment on the case of non-compact TOP 2-manifolds, which need extra information about "how they go to infinity" to be fully classified (the "space of ends" invariant) as well as the case of bounded TOP 2-manifolds, which, on top of it all, need the number of boundary components to be fully classified. Finally, we finish with the theorem by Rado that states that the DIFF structure on TOP 2-manifolds is unique up to diffeomorphisms. Thus, a full classification at the TOP category is also a full classification at the DIFF category.