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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.

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    $\begingroup$ For complex bundles and complex line bundle I think this is used in the construction of branched covers. $\endgroup$
    – Thomas Rot
    Nov 14, 2017 at 14:07
  • $\begingroup$ @ThomasRot for complex bundles is also looks like the equivalence between Cartier and Weil divisors $\endgroup$ Nov 16, 2017 at 17:49

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