A rank $r$ real vector bundle $p : E \to B$ is said to be orientable if there is a Thom class $\tau \in H^r (D(E),\partial D(E) ; \mathbb{F})$ (where $D(E)$ is a unit disk bundle and $\mathbb{F}$ a field) if its restriction to each fibre $(D(E)_b,\partial D(E)_b)$ for $b \in B$ is a generator of the group $H^r ( D(E)_b, \partial D(E)_b ; \mathbb{F})$.
If we swap the cohomology groups for homology groups (by Poincare duality) does that given an equivalent definition?
Thank you.