The following fact looks like well-known. I can prove it myself, but is there any reference to proof?
Let $ M$ be a $ n$ -manifold, $ C\subset M$ 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 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 gomologous to $ C$ , but here exact equality is required.)
What classical book or artical on obstruction theory contains assertions of this kind? I do not need a proof but only a reference.
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