On page 47 of Hatcher's Alg Top, he says
Letting x vary, these radial segments then trace out a copy of the mapping cylinder $X_m$ in the first solid torus.
Why is this the case? My supposition is that the core circle of the torus is the $S^I$ factor in the $S^1\times I$, but then I'm not sure how to comprehend the fact that there are three copies of $I$ attached to $S^1$. Or for that matter to make sense of how this sentence relates to the quotient space construction (of a mapping cylinder).
I'm sorry if this turns out to be a trivial point. While there's an old almost identical question, I need yet more clarification.
I'll be grateful for a response.
Best Answer
I suggest seeing it this way: in a mapping cylinder $C_f$ of $f: X \to Y,$ above each $y \in Y$ hangs a segment for each $x \in f^{-1}(y),$ and these segments vary continuously in $C_f$ when $y$ varies in $Y.$ (This follows from staring at the mapping cylinder construction intently.) But this is exactly the case here: for instance, in the picture below that Hatcher offers there are three bold segments going from the center, and the central point is going to move along the circle (perpendicular to the screen). Accordingly, in the mapping cylinder of $S^1 \xrightarrow{\times 3} S^1$ you have three segments sticking from each point of the codomain $S^1,$ and their ends form the domain $S^1$.
[Aside: you would've increased your chances for an answer a lot had you provided the context and for example the above picture. As given, the question is understandable only to those who read this part and still remember something.]