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I am trying to glue a handle to a torus with two open discs removed. But I fail because I can't parametrize the torus. My try:

Parametrize the cylinder (handle ) $C$ as $(h,\theta)$ and the torus with discs removed $T'$ by $(\varphi, \theta)$. It is very difficult to specify the suitable equivalence relation on $C \sqcup T'$. Can you suggest parametrizations that it is easier? Thank you for correcting me.

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Just realize that disk is a same thing as rectangle and similarly for the base of the cylinder. The rectangle can then be specified easily in the coordinates for $T'$ you picked. – Marek Jan 18 at 10:26

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