So simple!! You can either:

- cut two holes and glue a handle. This is easy to visualize as it can be embedded in $R_{3}$: you just get a Torus, then a double torus, and so on
- cut a single hole and glue aMöbius strip in it. Keep in mind that this is possible because the Möbius strip has a single boundary just like the hole you just cut. This leads to another infinite family that starts with:

A handle cancels out a Möbius strip, so adding one of each does not lead to a new object.

You can glue a Mobius strip into a single hole in dimension larger than 3! And it gives you a Klein bottle!

Intuitively speaking, they can be sees as the smooth surfaces in N-dimensional space (called an embedding), such that deforming them is allowed. 4-dimensions is enough to embed cover all the cases: 3 is not enough because of the Klein bottle and family.

- Generalized Poincaré conjecture | 282, 526, 6
- Homotopy | 0, 526, 7
- Topology | 72, 2k, 54
- Calculus | 17, 8k, 203
- Mathematics | 17, 28k, 633
- Ciro Santilli's Homepage | 262, 218k, 4k

- Classification | 107
- Generalized Poincaré conjecture | 282, 526, 6
- Klein bottle | 29
- Real projective plane | 624, 624, 2
- The beauty of mathematics | 391, 832, 8