I recently did an exam of Basic Algebraic Topology and could not get my head around this question.
I have this map $f:\Bbb S^1\to\Bbb S^1\times\Bbb S^1$, $f(z)=(z,z^q)$, for $q\ge1$. I could calculate the fundamental group for $X = (\Bbb S^1\times\Bbb S^1)\cup_f e^2$, and if I'm not mistaken it is $\Bbb Z$.
But then they asked me to prove that any map $g:X\to\Bbb S^1\times\Bbb S^1$ is homotopic to a constant map. I tried to use the lifting properties of the covering maps, and tried to check $\operatorname{im}(g_*)$, where $g_*:\pi_1(X)=\Bbb Z\to\pi_1(\Bbb S^1\times\Bbb S^1) = \Bbb Z \times \Bbb Z$, in order to check if $\operatorname{im}(g_*)\subset \operatorname{im}(p_*)$, for $p_*$ the induced map of a covering space. Is it true that $\operatorname{im}(g_*) = \Bbb Z$ or must it be a subset of $\Bbb Z \times \Bbb Z$? Also, which covering map could I choose to do this approach?
The last question asked me to get all covering spaces of $X$, for $q=4$ up to homotopy equivalences. I don't get why did they need to fix $q$, if we could easily do it with the classification of covering spaces with a base point. But then which covering spaces do I choose for the subsets of $\Bbb Z$ (again, if I'm not mistaken, they are $m\Bbb Z$, for $m\in \Bbb Z$ and $\{0\}$)?
Edit: I know that the question in hand is wrong. Thanks to user881318 for the answer and to Кряжев Арсений for the comment!
I also wanted to correct that the last question did not ask of computing the covering spaces of $X$, but rather say how many covering spaces $X$ had. This is much simpler, since it has as many covering spaces as subgroups $\pi(X,x_0)$ has (which is infinitely many, considering that $\pi(X,x_0) = \Bbb Z$, and the subgroups are $m\Bbb Z$, for $m\in \Bbb Z$ and $\{0\}$).
Every map $X \to T^2$ arises as a pair of a map $g: T^2 \to T^2$ and a null-homotopy of $fg$.
The map $g: T^2 \to T^2$ is determined up to homotopy by its induced map on $\Bbb Z^2$, which is given by some 2 x 2 integer matrix $A$.
The assumption that $fg$ is null-homotopic, then, is the assumption that $$A \begin{pmatrix} 1 \\ q \end{pmatrix} = 0;$$ this is precisely the necessary condition to extend $g$ to a map $\overline g$ from $X$.
This condition is simple enough. It means that $$A = \begin{pmatrix} nq & -n \\ mq & - m \end{pmatrix}$$ for some pair of integers $n,m$.
So take such a nonzero matrix $A$ and write $\overline g_A: X \to T^2$ for a choice of extension of the corresponding $g_A$ to $X$. Then $\overline g_A$ is not null-homotopic, because its restriction to $T^2$ --- $g_A$ --- is not null-homotopic, because its induced map on fundamental groups --- $A$ --- is nonzero.
So the premise of your exam question is either wrong or misstated.