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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"

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What is your question? What you quote is not a question, it's a definition (of $X$). –  Arturo Magidin May 30 '11 at 3:20
    
@Arturo. Thank you for your remark. I am supposed to compute the homology of the quotient space. –  El Moro May 30 '11 at 3:31
    
Could you please add that to the question to clarify it? It's also not clear to me whether you are having trouble conceptualizing what $X$ is, or just with computing the homology of $X$. –  Arturo Magidin May 30 '11 at 3:33
    
@Arturo I want to compute the homology .I would be grateful to you if you also could help me understand how f works on homology –  El Moro May 30 '11 at 3:49
    
please edit your question to that it becomes a question. Not in the comments: in the actual question. –  Mariano Suárez-Alvarez May 30 '11 at 4:17
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1 Answer

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.

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