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Whenever we have a continuous map of topological spaces $f: X \to Y$ and a sheaf $\mathcal{F}$ on $Y$ (of abelian groups for example), we get an induced pullback map $$ H^n(Y, \mathcal{F}) \to H^n(X, …

Let $p: E \to B$ be a locally trivial fibration with fiber $F$. If necessary, suppose that $B$ is simply connected. Suppose that the Serre spectral sequence leaves the term $H_p(B, H_q(F, \mathbb{Q})) …

The Wikipedia page for Borel-Moore homology states that for a locally compact set $X$ and a closed subset $Z$, if we write $U = X \setminus Z$ we have the following long exact sequence $$\cdots \to H^ …

Consider a locally trivial fibration $f: E \to B$ with fiber $F = \mathbb{C}^n$. The Leray spectral sequence with compact supports is $$ E_2: H^p_c(B, \underline{H^q_c(F)}) \implies H^{p+q}_c(E). $$ S …

Let $f: X \to Y$ be a locally trivial fibration between locally compact spaces with fiber $F$. It is well known that for a constant sheaf $A_X$ on $X$, the higher direct images $R^n f_* A_X$ are local …

With the information added in the last update, the result is already proved once you realize that the unit of the adjunction $$ id \to a_{B,*} a_B^* $$ is actually an isomorphism on the category of ab …

I am looking for a reference for the following result: If $X$ is an algebraic variety and $$X = T_n \supset T_{n-1} \supset \cdots \supset T_{-1} = \varnothing$$ is a stratification (edit: filtration) …

Let $L \subset \mathbb{C}^n$ be a hyperplane and let $i:L \to \mathbb{C}^n$ be the inclusion. Since $i$ is proper, we have induced maps $i^*: H^k_c(\mathbb{C}^n) \to H^k_c(L)$, and these maps are zero …