Homotopy Equivalence of Punctured Tori 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!
 A: 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 :)
A: 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.
A: 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.
