Sphere being non contractible: physical proof?

Question

I was wondering what would be a "physicist" argument to the non contractibility of the sphere?

I am pretty sure I read something about it on internet. It used the flux of a vector field and a triple integral of a point being strictly positive, unfortunately I couldn't find it.

Answer 1

Here is an analytic proof due to John Milnor of the Brouwer's fixed point theorem, from which it follows that the sphere is not a retract of the disk.

See this answer for why this is equivalent to noncontractibility.

Ultimately, this is just an analytic proof for a basic fact in algebraic topology, and the rest is using the theory as usual, I'm not sure exactly what you were looking for though.