Question

I have always wondered about applications of Algebraic Topology to Physics, seeing as am I studying algebraic topology and physics is cool and pretty. My initial thoughts would be that since most invariants and constructions in algebraic topology can not tell the difference between a line and a point and R4 so how could we get anything physically useful?

Of course we know this is wrong. Or at least I am told it is wrong since several people tell me that both are used. I would love to see some examples of applications of topology or algebraic topology to getting actual results or concepts clarified in physics. One example I always here is "K-theory is the proper receptacle for charge" and maybe someone could start by elaborating on that.

I am sure there are other common examples I am missing.

Answer 1

First a warning: I don't know much about either algebraic topology or its uses of physics but I know of some places so hopefully you'll find this useful.
Topological defects in space

The standard (but very nice) example is Aharonov-Bohm effect which considers a solenoid and a charged particle. Idealizing the situation let the solenoid be infinite so that you'll obtain with a line removed.

Because the particle is charged it transforms under the gauge theory. More precisely, its phase will be parallel-transported along its path. If the path encloses the solenoid then the phase will be nontrivial whereas if it doesn't enclose it, the phase will be zero. This is because and note that vanishes outside the solenoid.

The punchline is that because of the above argument the phase factor is a topological invariant for paths that go between some two fixed points. So this will produce an interference between topologically distinguishable paths (which might have a different phase factor).
Instantons

One place where homotopy pops up are Instantons in gauge theories.

Specifically, if you consider a Yang-mills theory in (so this means Euclidean time) and you want the solution (which is a connection) to have a finite energy then its curvature has to vanish at infinity. This allows you to restrict your attention to (this is where the term instanton comes from; it is localized) and this is where homotopy enters to tell you about topologically inequivalent ways the field can wrap around . Things like these are really big in modern physics (both QCD and string theory) because instantons give you a way to talk about non-perturbative phenomena in QFT. But I am afraid I can't really tell you anything more than this. (I hope I'll get to study these things more myself).
TQFT

Last point (which I know nearly nothing about) concerns Topological Quantum Field Theory like Chern-Simons theory. These again arise in string theory (as does all of modern mathematics). And again, I am sorry I cannot tell you more than this yet