Homotopy Equivalence of Punctured Tori

Question

I was told recently that if I take a 2-torus (genus 2) and remove 1 point, then this is homotopy equivalent to a torus with 3 points removed. This may be really easy but I don't see it.

Thank you!

Answer 1

A surface minus a finite number of points is homotopy equivalent to a bouquet of circles, and two bouquets of circles are homotopy equivalent iff they have the same number of circles.

This two observations and a little picture to see how many circles are involved in your example should do it :)

Answer 2

First observe that the fundamental groups of both spaces are isomorphic (both are free groups with 4 generators). Both universal covers are contractible (they are 2 dimensional non compact spaces). Hence both spaces are homotopy equivalent to the classifying space of the free group with 4 generators.

Answer 3

If you can see it for the torus and the sphere, then adding genus should be no problem.

So picture an ordinary torus, a hollow doughnut (not a donut, that's a sphere, once you've taken out the jam). Puncture it. Now use that hole to drag back the surface. You'll end up with two rings, joined at a point. That point is flexible so allow the rings to hang from it, but imagine a little repulsion between them so that they hang slightly apart. From these two rings, you can grow a sphere: each ring gets filled in (apart from a centre point) and the space between the two rings gets filled in as well (again, apart from one point). Do all that and, voila!, a sphere with three holes. Or a donut that's had the jam scooped out by three kids simultaneously.

To add genus, just do the following: instead of allowing your doughnut to retract back to two infinitely-thin rings, stop a little short so that they are strips. Or at least, that one of them has a little bit of thickness somewhere (preferably away from the point of contact with the other). Now add a handle to this thickened ring. That handle can stay there throughout the whole process, innocently adding genus to everything around, so the doughnut becomes doughnutty, and the donut becomes a doughnut.