Are there any interesting connections between Game Theory and Algebraic Topology?

One example is in the concept of a Nash equilibrium, whose existence can be proved using various (topological) fixed point theorems. (Google "nash equilibrium proof" for a wide variety of examples... the main topological machinery that comes up is the Kakutani fixed point theorem or the plain-vanilla Brouwer theorem.)

Fixed point theorems come up in many other, similar settings; you see them a lot in certain kinds of mathematical economics. The overall intuition is that given a current set of known player strategies, people will generally want to change to better strategies. This defines a function on the "space of strategies". A fixed point of this function is interesting because it indicates an "equlibrium" of strategies that are in a sense optimal. Often topology is the best way to prove a fixed point exists.

There's a neat paper by David Gale on the connection between Brouwer's fixed point theorem and the game of Hex. The fixed point theorem is 'equivalent' to the theorem that the game cannot end in a tie.

The Banach-Mazur game is an example of a game in a topological setting. There are various other games of this nature which are mostly related to foundational questions in point-set topology.

I'm afraid I can't think of a meaningful connection between game theory and algebraic topology. I think that real algebraic geometry plays a part in the study of various types of equilibria in a game-theoretic context, but that's about alll I can say about this.