I have an excercise that I kind of dislike:
Given $T-\{p,q\}$ where $T = S^1 \times S^1 $ and $p,q \in T$ two different points, I am supposed to find a simple homotopy equivalent space by imagination. So, no explicit formulas are necessary, but I just don't understand how to start this excercise. I mean, what is the first thing to do now?
Similarly I have to do this for three points missing in $RP^2$ and two points missing in the Klein bottle.
I am really stuck. How would you start with this excercise?
I would start by thinking of the torus as a square, with two pairs of sides identified.
If the exercise were to find an equivalent for the once punctured torus, I would think of a punctured square, which is obviously equivalent to a circle. But considering the identifications, the punctured torus is homotopy equivalent to two circles that share a point.
This may give you a direction to your actual exercise.