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 $\mathbb{R}^4$ 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.
Best Answer
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 ${\mathbb R}^3$ with a line removed.
Because the particle is charged it transforms under the $U(1)$ 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 $$\phi \propto \oint_{\partial S} {\mathbf A} \cdot d{\mathbf x} = \int_S \nabla \times {\mathbf A} \cdot d{\mathbf S} = \int_S {\mathbf B}\cdot d{\mathbf S}$$ and note that $\mathbf B$ 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 ${\mathbb R}^4$ (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 $S^3$ (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 $S^3$. 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.