Stuck in Space - Part I

Question

I've been transported through a wormhole to a new planet in a new universe.

Standard stuff, really.

But something is... off about this planet. Sometimes, walking around it, I just feel like I'm going in the wrong direction entirely. And it gets weirder. When I first arrived, I decided to make the best of the situation and stick a flag in the ground, claiming the land as my own. That flag became my home base, but one day during my exploring I got particularly lost and found the bottom of the flagpole sticking out of the ground. I don't remember how I got back that night, but I eventually stumbled back to the original flag - absolutely untouched.

Thanks to some recent developments, I've been able to contact a team of mathematicians on the other side of the wormhole to help resolve my problems. It seems the communication is one-way, though, because they are now stuck in the wormhole trying to get to my side. The only thing that got through seems like a piece of one of their notes:

  <
v   v  Z,Z(F2),0,...
  > 

What is this planet, and what is this universe? Where the hell am I?

Answer 1

I think you are on

a Klein bottle.

The note shows

on the left, a diagram showing how to make one by identifying edges of a rectangle, and on the right the homology groups of the Klein bottle (though the notation is a little peculiar).

And

the Klein bottle is nonorientable, which is how you were able to find yourself on the "wrong side" of it after some wandering around.

As for the universe,

it's hard to say much with confidence, but your ambient space can't be R^3 because you can't embed a Klein bottle in that. Perhaps somewhere with more dimensions, or perhaps the universe itself is nonorientable.