Suppose $X,Y$ are locally compact Hausdorff spaces and $f:X\to Y$ is a topological submersion of relative dimension $n$. By this we mean that for all points $x\in X$, there exists an open neighborhood $U\subset X$ and an open embedding $U\hookrightarrow\mathbb R^n\times Y$, commuting with the projection to $Y$. In particular for each $y\in Y$, the fibre $X_y:=f^{-1}(y)\subset X$ is naturally a topological manifold of dimension $n$.

Now, fix a commutative ring $k$. Let $\underline k_Y$ denote the constant sheaf on $Y$ with stalk $k$ and let $\mathfrak o_{X/Y}$ denote the relative orientation local system of $f$, i.e., its stalk at a point $x\in X$ is given by $(\mathfrak o_{X/Y})_x=H_n(X_y,X_y\setminus\{x\};k)$, where we have set $y:=f(x)\in Y$.

Suppose $E$ is a local system of $k$-modules on $Y$. I would then like to define a pushforward map $\int_{X/Y}:H^{\bullet+n}_c(X,\mathfrak o_{X/Y}\otimes_k f^*E)\to H^{\bullet}_c(Y,E)$ which is morally given by "integration along the fibre". **What is a reference where this map is constructed and some of its basic properties (e.g. functoriality, compatibility with Mayer-Vietoris sequence etc) are proved?** More generally, what are some references which treat a relative version of Poincare duality?