Pushforward in Compactly Supported Cohomology

Question

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?

Answer 1

You get your integration map as follows. There is a map of complexes of sheaves on $Y$: $$ f_!f^! E \to E$$ given by the counit of the adjunction (where all functors are derived). This gives rise to your map after applying $\Gamma_c$.

To see that applying $\Gamma_c$ gives your map you need to identify $f^!E$ with $\mathfrak o_{X/Y}[n] \otimes f^\ast E$ (where $[n]$ denotes a degree shift). And this is because $\mathfrak o_{X/Y}[n] = f^!\mathbf Z_Y$ and because $f^!$ satisfies the identity $f^!(F \otimes G) = f^!(F) \otimes f^\ast(G)$. In particular the "relative version of Poincaré duality" is simply the calculation of $f^!\mathbf Z$ where $f$ is a submersion. The properties I used above are in the standard references like Iversen and Kashiwara-Schapira. This is a bit sketchy so let me know if I should elaborate on something.