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

Question

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.

Answer 1

Hint: show that both spaces deformation-retract to a wedge of two circles.

Answer 2

Look at the polygonal representation of two spaces. Now removing a disc from the middle, the rest of the space will be deformation retract into the boundary, which is nothing but wedge of two circles. (Just draw the picture of polygonal presentation, you can actually see what is happening.)