Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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

share|cite|improve this question
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

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.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.