My professor seems to randomly use the symbols [itex]\approx[/itex], [itex]\cong[/itex] and [itex]\simeq[/itex] to denote either homeomorphic, isomorphic or homotopic. So, what is the official symbol for each?
I don't think there's a universal convention for the first and the third, but in group theory "isomorphic to" is denoted by [itex] \simeq [/itex].
Isomorphic is more frequently [itex]\cong[/itex], though the latex tag for it is cong. The symbol Dexter gave is actually in my experience likely to beused for homotopic, though I also see [itex]\sim[/itex] a lot as well.
I have never used a text in which the author used a symbol for these concepts without first defining it.
That's what my post was meant to convey: there is no standard convention and so each author sets up his text to so that the symbol he likes most is used most frequently.