This question is on an old qualifying exam for my institution:
Let $Y$ be a space and $f:Y\to Y$ a self-mapping. Let $X$ be the mapping torus of $f$ (i.e. the space obtained from $Y\times I$ by identifying $(y,1)$ and $(f(y),0)$ for $y\in Y$). Prove that $H_1(X;\Bbb Z)\cong H_1(Y;\Bbb Z)/\mathrm{im}(\mathrm{id}-f_*)$ where $f_*:H_1(Y;\Bbb Z)\to H_1(Y;\Bbb Z)$ is the induced map.
Now, the reason I am confused is because if you take $f$ to be the identity, then $X$ is just $Y\times S^1$ and in particular if we take $Y$ to be, say, a point then $X=S^1$ and we get
$$\Bbb Z=H_1(X;\Bbb Z)=H_1(Y;\Bbb Z)/\mathrm{im}(\mathrm{id}-f_*)=0.$$
Is there some flaw in my logic or is the question missing some assumption?