define orientation of quotient space

Question

How can one expand the definition of orientation of vector spaces to corresponding quotient space? For example, if I am given two vector spaces V and W, and I know the definition of orientation for both of them (I am free to symbolize these orientations, such as a function $O_1$:Bases(V)->±1 for V, and $O_2$:Bases(W)->±1 for W), how can I define an orientation of quotient space W/V?

Answer 1

When you want to consider the quotient $W/V$, this makes only sense if $V$ is a subspace of $W$, that is $V \subseteq W$. To orient the quotient, you can do the following: Given a base $(w_1 + V, \ldots, w_r + V)$ or $W/V$, lift it to $W$, that is, consider $(w_1, \ldots, w_r)$. Now take a base of $V$, say $(v_1, \ldots, v_k)$ and define $$ O_{W/V}(w_1 + V, \ldots, w_r + V) := O_1(w_1, \ldots, w_r, v_1, \ldots, v_k)O_2(v_1, \ldots, v_k) $$ (You have of course to check that this is well-defined and does not depend on any of the choices made).