# Pushforward in Compactly Supported Cohomology

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?

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.
• Thanks! Could you give a more precise reference for the computation $\mathfrak o_{X/Y}[n] = f^!\mathbb Z_Y$? The case $Y=pt$ seems to be done explicitly in Kashiwara-Schapira, but I'm not able to find the general case. – Mohan Swaminathan Nov 23 '19 at 16:39
• Yes, specifically Prop 3.3.6 (i), (ii) which does the computation in the case $Y=pt$. – Mohan Swaminathan Nov 23 '19 at 17:59
• Look at Equation (3.3.1) in the proof of Prop 3.3.2 and the following lines to see how to go from $Y=pt$ to the general case. I agree it's not stated explicitly in the way you want it, though. – Dan Petersen Nov 23 '19 at 22:06