In basically every introduction to Morse theory, we see the example of a height function on a torus. Morse theory then predicts that a torus minus a disk is homotopy equivalent to a cylinder with a 1-cell attached (see the figure below). I cannot see the homotopy equivalence. Can someone explain it to me?
They're both $S^1\vee S^1$. The figure on the right is homotopy equivalent to the one on the left by continuing to stretch the hole down until you just have a basket-handle formed by the top of the "donut hole."
Alternatively, you can form a torus by rolling up a sheet of paper and gluing the ends together. Either of these figures (as well as $S^1\vee S^1$) can be obtained by cutting out a window in the sheet of paper first.