Let $Y\subset X$ be a codimension $k$ proper inclusion of submanifolds. If we choose a coorientation of $Y$ inside of $X$ (that is, an orientation of the normal bundle), then we get a class $[Y]\in H^k(X)$. If $X$ and $Y$ are oriented, then $[Y]$ may be defined as the fundamental class of $Y$ in the Borel-Moore homology of $X$, which is isomorphic to the cohomology of $X$. What is the simplest definition of $[Y]$ in the general case (where $X$ and $Y$ are not necessarily oriented)?


Note that a simple generalization of this question would be to ask how to define the pushforward in cohomology along a proper oriented map. (Then $[Y]$ would simply be the pushforward of $1\in H^0(Y)$.) I would be happy to know the answer to this more general question, but I asked the simpler version to be as concrete as possible.



Cohomology classes represented by submanifolds
The easiest definition is via the Pontrjagin--Thom construction (I think). Let $N$ be a tubular neighbourhood of $Y$, isomorphic to the normal bundle. Let $X'$ be the space obtained from $X$ by collapsing the complement of $N$ to a point. Then $X'$ is isomorphic to the Thom space of the normal bundle, and if the normal bundle is oriented then there is the Thom isomorphism:
$$ H^\bullet(Y,\mathbf Z) \cong H^{\bullet+\dim X - \dim Y}(X',\mathbf Z).$$
Composing with the natural $H^\bullet(X',\mathbf Z) \to H^\bullet(X, \mathbf Z)$ gives the map you want.


