# Is torus w. disc removed homotopic to Klein bottle w. disc removed?

I know that homeomorhic spaces are homotopic, but am not sure if this applies, since I think they are not homeomorphic due to orientability.

I know $f$ and $g$ are homotopic if they represent: X$\rightarrow$Y, and there exists Homotopy map: $H: X \times [0,1] \rightarrow Y$, with: $H(x,0)=f(x)$ and $H(x,1)=g(x)$

So $X$ is a torus with 2-disc removed, $Y$ is the Klein bottle with 2-disc removed, but I am not sure how to apply the equation in practice.

It would be great if someone could help, and then I can practice more questions.

• Are you familiar with the fundamental polygons of those spaces? This should enable you to prove that both spaces are homeomorphic to a wedge sum of two circles. – Ayman Hourieh Feb 9 '16 at 21:22
• Two "different" knots are homeomorphic but not homotopic. – ajotatxe Feb 9 '16 at 21:24
• @ajotatxe, I cannot tell what the connection is between that and the question! – Mariano Suárez-Álvarez Feb 9 '16 at 21:25
• @thinker I've posted an answer, please have a look and let me know if you have doubts – Anubhav Mukherjee May 20 '16 at 13:42

• @Anubhav.K, the OP is not removing two discs, he is removing one $2$-disc... – Mariano Suárez-Álvarez Feb 9 '16 at 23:40