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Is $S_1 \times S_1 \sqcup_{(x,y) \sim (x,x+y)} S_1 \times S_1$ homotopy equivalent to $S_1$?

I do not think it is. My motivation is in proving that the fiber of a map in a certain math overflow answer by a senior professor, who I don't want to name, is wrong.

Here I am letting the the torus be $[0,1]_A/\{0,1\} \times [0,1]_B/\{0,1\}$. This identification identifies the based homology class $A+B$ in the second copy of the torus with the homology class $A$ in the first copy of the torus.

Do you have any ideas of showing how this is not $S_1$?

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    The map $(x,y)\mapsto (x,x+y)$ is a homeomorphism on the torus $S^1\times S^1$, thus the resulting space is just obtained by gluing two torus via a auto-homeomorphism, thus a torus, clearly not homotopic to a circle $S^1$. – 2017-02-02
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    Thanks. Sorry for being thick. – 2017-02-02

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