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Can anyone suggest at least verbally how it is that a torus is constructed from two cells? (Total beginner to topology, been away from math as a whole for awhile...) Thanks for any insights!

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    @EricO.Korman From the video: To clarify, what is the 2-cell? My understanding is that an n-cell is homeomorphic to an n-ball. (Is that correct? Is this the precise definition?) So is the 2-cell the original rectangle (which is homeomorphic to a 2-ball?) and by bending the rectangle as done in the video, its boundary is being "attached" along two blue arcs (the 1 cells) and then joined at a single point (the 0 cell)?2016-02-22

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The torus can be made from a single 0-cell, two 1-cells and one 2-cell. The 1-cells are attached along their boundary to the 0-cell, making two circles identified at a point. Then the 2-cell is attached to this 1-skeleton, as illustrated in the comments above.

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The point of view offered by Jim is good because it can be generalized: if you take any surface with $n$ holes (say a connected sum of $n$ tori), it has a CW-complex structure made by a single 0-cell, $2n$ 1-cells 'wedged' to the 0-cell and a single 2-cell applied on the 1-cells. It is page 5 of Hatcher's Algebraic Topology.