smoothManifold(d, v, i, S)
This function accesses a database of all small triangulated 2 or 3manifolds with at most ten vertices. The enumeration of these abstract simplicial complex follows Frank H. Lutz's classification in "Triangulated Manifolds with Few Vertices: Combinatorial Manifolds", arXiv:math/0506372v1. Alternative formats, further references, and more details may be found at the manifold page.
There is a $1$ surface with four vertices, $1$ with five vertices, $3$ with six vertices, $9$ with seven vertices, $43$ with eight vertices, $655$ with nine vertices, and $42,426$ with ten vertices. There is $1$ threefold with five vertices, $2$ with six vertices, $5$ with seven vertices, $39$ with eight vertices, and $1,297$ with nine vertices.






Frank Lutz also includes some information of the topological type of the manifolds on his webpage. The 2sphere, 2torus, real projective plane and Klein Bottle can all be accessed from the database. Here are the smallest instances which appear in the database for real projective plane and the Klein Bottle (note that our index begins counting at zero instead of one):


From Theorems 6.3 and 6.4 in Chapter one of Munkres' Algebraic Topology, both real projective plane and the Klein Bottle should have zero homology except in the first position.

