# How is topology related to physics?

Topology has many occurences in physics like topological insulators, topological quantum computing etc. But what is confusing me is that topology is this mathematical theory that studies the behaviour of spaces using transformations that are invariant under local perturbations. At least that is how I see it I'm not an expert in topology. It seems to me that physics only adapted the 'invariant under local perturbations' part and uses this to define topological invariants. So is this really related to topology or did we just steal the name? Can we use the machinery of topology for these topological invariants? Can topology help find these topological invariants?

• I'm unclear what you're asking. Are you asking if topological invariants belong to the study of topology? – jacob1729 Oct 15 at 13:56
• You may regard a physical theory as a "procedure" to calculate some "physical quantities" of interest X. If the values of $X$ are independent on the ambient space metric, then you have a "topological" physical theory: the value of X will depend on the overall shape of the ambient space (usually the spacetime, but this stuff is applied also in Stat Mech contexts), not on its deformations. So, to me it looks like this stuff "is really related to topology" :) – Quillo Oct 15 at 13:57
• @jacob1729 I guess so. To me it seems they are only loosely connected. But I believe this is probably because I understand it wrong – AccidentalTaylorExpansion Oct 15 at 16:30