Algebraic Topology Ideas

Random number: 27

8th February 2017

It's been a while; Algebraic Topology is super hard its taking up all my time. I've had a couple ideas floating about my head.

Wouldn't it be interesting if we could somehow find a pointed topological space \(X,x_0\) such that there is an injection \(i\) from the set of all convergent infinite sums of real numbers into the set of all paths centered at \(x_0\) such that two sums give the same result if and only if the two corresponding paths are homeomorphic? That would be cool; because it seems like a nice way of converting a problem of sums into a problem of topology. That said; it also sounds a bit completely useless; because it probably doesn't make the problem of finding a sum any easier. Or maybe it does. I don't know.

This is by no means an original idea; but I do hope also that we live in a non-orientable universe; such that if I travel on a particular loop I could come back to Earth and in everyone else's perspective I would have changed from right-handed to left-handed.