Y on a Closed Surface

As shown in the "Hex on a Torus" topic, it is possible to generalize Hex on a closed surface (a board with no boundaries) with various winning conditions:

If the winning conditions are:

Red: A vertical loop

Blue: A horizontal loop

This minimum setup works:

(Torus)

http://img59.imageshack.us/img59/7420/torustohex.png

If the winning conditions are:

Red: Prevent blue from making any loop

Blue: Any loop

This minimum setup works:

(Torus)

http://img33.imageshack.us/img33/1672/torustohexalt.png

(Discovered by Jonathan Rydh)

Now, my new question is this: Is it possible to generalize Y on a closed surface?

I am almost positive that it cannot be generalized on a torus. However, what about a different closed surface, such as a sphere, Klein bottle, or real projective plane?

I'm not yet convinced for the Hex on torus variant...

"I'm not yet convinced for the Hex on torus variant..."

Just like one can play Hex with a Y ruleset:

http://en.wikipedia.org/wiki/File:9x9-Y-Board-to-Hex.svg

Hex can be played with on a torus with those starting setups.