Quite often in algebraic topology I see the word "preferred" come up (online) but I have not seen a definition for it anywhere. As an example the solution for the problem here:
Covering space action on an orientable manifold $M$ implies $M/G$ orientable (Hatcher)
uses the word four times, and the question asker used it once.
"You have to try to push down the orientation μ to M/G. To do this you have to use, that there are no orientation reversing covering homeomorphisms, by noting that the preferred generators of any orbit in the covering gives you the same preferred generator at the image point in M/G. For this just consider the commutative diagram of local homology groups, corresponding to a covering translation of G together with the projection. Now note that under a orientation preserving homeomorphism, preferred generator is mapped to preferred generator."
I am learning algebraic topology out of Hatcher and although he uses the word three times (in the appendix to section 4) he does not define it anywhere.
I apologize if this is an inappropriate place to be asking such a question.
What's going on here is that an orientation $\mu$ on a topological $n$-manifold $M$ is determined by a function which assigns to each $x \in M$ one of the two generators of the infinite cyclic local homology group $H_n(M,M-x;\mathbb Z)$ (subject to a compatibility condition which I won't go into). You could refer to that generator with notation such as $\mu_x \in H_n(M,M-x;\mathbb Z)$, or you could refer to it in words as "the preferred generator at $x$".