Is it true that gluing can be realized two ways? In the example of gluing two cylinders to obtain a torus one can imbed each cylinders into the torus like this:
A point $(\cos \phi , \sin \phi, \theta)$ in $C_1 = S^1 \times [0,1]$ maps to $x = \cos(2 \theta)(R+r\cos(\phi))$, $y=\sin(2 \theta)(R+r\cos(\phi))$, $z=r\sin(\phi)$ and $(\cos \phi , \sin \phi, \theta)$ in $C_2 = S^1 \times [0,1]$ maps to $x = \cos(\pi + 2 \theta)(R+r\cos(\phi))$, $y=\sin(\pi + 2 \theta)(R+r\cos(\phi))$, $z=r\sin(\phi)$. These maps define an imbedding into the torus in $\mathbb R^3$. This is a gluing where the boudaries of the cylinders are identified.
A different way of realizing the gluing: Set $X = C_1 \times \{1\} \cup C_2 \times \{2\}$. Define $f: \partial C_1 \times \{1\} \to \partial C_2 \times \{2\} $ as $(s,1,1) \to (s,1,2)$ and $(s,0,1) \to (s,0,2)$. Then define $x \sim y \iff f(x) = y$ and $T = X / \sim$.
Is the first the same as the second but with an imbedding i.e.: Does one have to make a quotient space in the first also (before imbedding)? Does gluing always mean take quotient?
