## FANDOM

561 Pages

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
$4\mathbb{T}^2$
$5\mathbb{T}^2$
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
$4\mathbb{RP}^2$
$6\mathbb{RP}^2$
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
$4\mathbb{RP}^2$
$6\mathbb{RP}^2$
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.