The final part of the exhibition presents relevant quotes and theorems on 2- and 3-manifolds.
We begin with the following quote from H. Poincaré:
"A geometry is no more true than another, it can only be more convenient..."
Then we present the statement of the classification of surfaces theorem, accompanied by explanatory figures.
Theorem (A.F. Moebius, 1869, and C. Jordan, 1866)
"All closed and orientable surface can be deformed to match a connected sum of
g torus to g ≥ 0."
Uniformization Theorem (H. Poincaré, P. Koebe, 1907)
"Any closed orientable surface admits one of the geometrical structures: Spherical (S2), flat (E2) or hyperbolic (H2).”
Then we show theorems for the case of 3-manifolds, which have been proved in 2002 by Perelman.
Conjecture (H. Poincaré, 1900)
"Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere."
Geometrization Theorem (Thurston, Perelman 2002)
"Every 3-manifold can be cut along surfaces and decomposed into three dimensional manifolds capable of forming one of the following geometries: S3, E3, H3, S2xR, H2xR, SL(2,R), Nil, Sun."
Finally, we highlight a notable quote from M. Morse that, about 70 years earlier, anticipated how these problems would be solved.
"Any problem of non-linear nature, involving more than one coordinate system or more than one variable, or whose structure is initially set globally, will probably require for their solution Topology and Group Theory considerations. In solving such problems, the classical analysis often appears as a local tool, integrated into the problem as a whole through Topology or Group Theory.”