FANDOM


Because every two-dimensional manifold is a connected sum of sum number of tori and some number of real projective planes, and there's a fancy relation where a torus and a RPP is actually just three RPPs, you can make a cool chart. Something like this (with the row and columns representing direct sums):

\mathbb{S}^2
Sphere

\mathbb{T}^2
Torus
2\mathbb{T}^2
Double torus
3\mathbb{T}^2
Triple torus
4\mathbb{T}^2
Quadruple torus
5\mathbb{T}^2
Quintuple torus
\mathbb{RP}^2
Real projective plane
3\mathbb{RP}^2
Dyck's surface
5\mathbb{RP}^2
Quintuple real projective plane
7\mathbb{RP}^2
Septuple real projective plane
9\mathbb{RP}^2
Nonuple real projective plane
11\mathbb{RP}^2
Undecuple real projective plane
2\mathbb{RP}^2
Klein bottle
4\mathbb{RP}^2
Quadruple real projective plane
6\mathbb{RP}^2
Sextuple real projective plane
8\mathbb{RP}^2
Octuple real projective plane
10\mathbb{RP}^2
Decuple real projective plane
12\mathbb{RP}^2
Duodecuple real projective plane
3\mathbb{RP}^2
Dyck's surface
5\mathbb{RP}^2
Quintuple real projective plane
7\mathbb{RP}^2
Septuple real projective plane
9\mathbb{RP}^2
Nonuple real projective plane
11\mathbb{RP}^2
Undecuple real projective plane
13\mathbb{RP}^2
Tredecuple real projective plane
4\mathbb{RP}^2
Quadruple real projective plane
6\mathbb{RP}^2
Sextuple real projective plane
8\mathbb{RP}^2
Octuple real projective plane
10\mathbb{RP}^2
Decuple real projective plane
12\mathbb{RP}^2
Duodecuple real projective plane
14\mathbb{RP}^2
Quattuordecuple real projective plane
5\mathbb{RP}^2
Quintuple real projective plane
7\mathbb{RP}^2
Septuple real projective plane
9\mathbb{RP}^2
Nonuple real projective plane
11\mathbb{RP}^2
Undecuple real projective plane
13\mathbb{RP}^2
Tredecuple real projective plane
15\mathbb{RP}^2
Quindecuple real projective plane

This was way more boring than I expected it to be, because after making this table I just realised that adding one cross-cap kills orientability so it's just going to be a whole bunch of crosscaps. Shrug emoji.

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