1. A familiar way to define a torus is to start with a square, say [0,1]x[0,1] in the plane, and identify corresponding points on the right and left edges, and also corresponding points on the top and bottom edges.
(I.e., identify (t,0) with (t,1), and identify (0,t) with (1,t), for all 0 ≤ t ≤ 1.)
Suppose that *in addition to those identifications*, we also identify (x,y) with (y,x) for all points (x,y) of the square (not just the edges). What familiar shape is the result? [] 2. Start again with a fresh, unidentified square [0,1]x[0,1]. Now:
* Identify (t,0) with (t+1/2 (mod 1), 1) for all 0 ≤ t ≤ 1,
and
* Identify (0,t) with (1, t+1/2 (mod 1)) for all 0 ≤ t ≤ 1.
What familiar shape is the result, topologically?
... spoiler space ... ... spoiler space ... ... spoiler space ... ... spoiler space ... ... spoiler space ... ... spoiler space ... ... spoiler space ... ... spoiler space ... ... spoiler space ... ... spoiler space ... ... spoiler space ... ... spoiler space ... ... spoiler space ... 1. Moebius strip. Clearly it's the same as taking half the square (cut along the diagonal x=y) and identifying its two orthogonal edges (in the way you _can't_ achieve by just folding it). Let's say we keep the northwest half of the square. Cut a little bit off the northwest corner; it's obvious that _this_ gives a Moebius strip (it's just the usual diagram deformed a bit). Un-cutting the corner is accomplished by identifying all the points of the cutting-off edge -- but that just means identifying an interval around the boundary of the Moebius strip, which obviously changes nothing. (Alternatively: it's obviously a compact surface with boundary, it's obviously nonorientable and has boundary a single circle, and we can easily calculate the genus via the Euler characteristic.) 2. Double torus. Divide the sides in half and think of them as the 8 sides of an octagon. We've identified opposite pairs of sides. Imagine doing only half of them; we get a torus with a square hole in it, whose opposite edges need identifying. Identifying one pair "closes up the hole" and gives us a torus with two holes whose boundaries are identified, and identifying those gives the second handle of the torus. (Alternatively: it's obviously a compact surface, it's obviously orientable, and even if you don't already know that identifying opposite edges of a 4g-gon yields the unique compact orientable surface of genus g, you can readily calculate the Euler characteristic.) -- g