TITLE: Cohomology classes represented by submanifolds
QUESTION [10 upvotes]: 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.

REPLY [3 votes]: This is an answer to your more general question about how to define $$f_\ast:H^\ast(X;\mathbb{Z})\to H^{\ast+\dim(Y)-\dim(X)}(Y;\mathbb{Z})$$
when $f: X\to Y$ is a proper oriented map.
For $\ell$ sufficiently large, there is an embedding $g: X\hookrightarrow \mathbb{R}^\ell$ which is unique up to isotopy. The map $f':X\to Y\times \mathbb{R}^\ell$ given by $f'(x)=(f(x),g(x))$ is then an embedding, and has a tubular neighbourhood $N$ as in Dan's answer. Note that the normal bundle $\nu_{f'}$ of $f'$ has dimension $\dim(Y)-\dim(X)+\ell$. There is a Pontrjagin-Thom collapse map $F:(Y\times\mathbb{R}^\ell)_+\to T(\nu_f')$, where $+$ denotes a one-point compactification. After making a standard identification, this gives a pointed map $F: \Sigma^\ell Y_+\to T(\nu_{f'})$, where $\Sigma^\ell$ denotes the $\ell$-fold reduced suspension.
Now we can define the pushforward as the composition
$$ H^\ast(X)\to \widetilde{H}^{\ast+\dim(Y)-\dim(X)+\ell}(T(\nu_{f'}))\stackrel{F^\ast}{\to}\widetilde{H}^{\ast+\dim(Y)-\dim(X)+\ell}(\Sigma^\ell Y_+) \to H^{\ast+\dim(Y)-\dim(X)}(Y).
$$
Here the first map is a Thom isomorphism and the last is a suspension isomorphism, and all coefficients are integers.