topology exam

[annathyst]

Good enough for partial credit. I didn't like Problem 2.
Close enough to being done that I can go to sleep and then hand it in in sure knowledge that I'll pass.

Night.

Comments

[http://users.livejournal.com/butter_fly_/]

explain the doughnut one to me later? i'm curious.

[billemon.livejournal.com]

The sides and top/bottom of the tic tac toe grid join up when you wrap it around a donut. So there are extra "diagonals" because

X--
--X
-X-

is actually the same as

--X
-X-
X--

. It would work better with pictures, I guess :)

[annathyst]

No, no, only one problem was tic-tac-toe and that was on a Klein bottle, not a torus. This one had to do with a punctured torus and showing homotopy equivalence to two circles joined at a point.

[billemon.livejournal.com]

OK, Klein bottle makes my brain hurt, but it's a single surface, so you could do the same thing, couldn't you? *flounders* This is why I did CS and not math lol.

[annathyst]

Not only is it a closed surface, you can get it by doing the same kind of thing you'd do with a square to turn it into a torus. You join the top and bottom edges like you would in a torus, but you join the left and right edges with a half-twist, so the bottom of one is joined to the top of the other and vice versa.

Don't try to do it, you can't in three dimensions :) But the resulting identification space is definitely a Klein bottle.

[slab]

Mmm... forbidden Klein bottle...

[billemon.livejournal.com]

Ahhhhhhh. but the twist puts you on the back of the board, and the next twist puts you back on the front the right way up :D:D you can just treat it the same .. </cheat>

[annathyst]

No, you can't. The board doesn't have a "back" - the square with the board drawn on it and edges joined up (in a space large enough for that to happen) IS the Klein bottle, not just a square chunk of it.

If you have this configuration:

X| |O
|O|
|X|X

that's a win for X in Klein bottle tictactoe, but not on the torus.

[annathyst]

That got mangled, but I think you see what I was getting at.

[billemon.livejournal.com]

I do - thanks. Topology always tied my head in knots, I'm afraid. That made sense though.