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I have to admit that I spent a while now thinking about the question below. I could see that the map f takes integers to integers keeping thus taking the vertices of $T^{2}$ to vertices of $R/Z$. I am thinking about Mayer vietoris but how no idea how to map $T^{2}$x$0$ to $T^{2}$x$1$ "in order to compute the homology of the quotient space"

Many thanks enter image description here

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    please edit your question to that it becomes a question. Not in the comments: in the actual question.2011-05-30

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Here is a hint for how to apply Meyer-Vietoris: Consider two open sets $U,V\subset [0,1]$, where $U$ contains the endpoints, and $V$ contains the middle. $U\cap V$ will have two connected components.

Now, consider U',V'\subset X given by U'=T^2\times U and V'=T^2\times V.. Because $f$ is an invertible linear map, both U' and V' are homotopy equivalent to $T^2$, U'\cap V' is homotopy equivalent to two copies of $T^2$.

Note that care must be taken, as $f$ will affect what the inclusion maps in the MV sequence look like. If you can determine how, the rest will fall out.