The following fact looks like it is well-known. I can prove it myself, but I would like to know of a reference which has a proof?
Let $M$ be an $n$-manifold, $C\subset M$ a codimension-1 closed submanifold, and $L\to M$ a real linear bundle. The only obstruction to constructing a nonzero section of $L$ lies in $H^1(M;\mathbb Z/2)$ and is called $w_1(L)$. Suppose $w_1(L)$ is dual to $C$. Then there is a generic section $s:M\to L$ such that $s(M)\cap M=C$.
(Obviously, if $w_1(L)=[C]$, then for every generic section, the submanifold of its zeroes is homologous to $C$, but here exact equality is required.)
What classical books or articles on obstruction theory contain assertions of this kind? I do not need a proof, just a reference.