I'm looking for concrete topological intuition for the derived pushforward.


Let $f:X\to Y$ be a continuous map. The derived pushforward $\mathbf Rf\_\ast$ takes a sheaf $F$ to the sheafification of the cohomology presheaf $V\mapsto \mathrm H^\bullet(f^{-1}V,F)$. When $f$ is the identity, the sheafification is zero for $n\geq 1$.


The sheafification of a presheaf $P$ can be constructed by taking mapping $PU$ to the equivalence class of families of sections of $P$ defined on an open cover of $U$, where we identify families which coincide on sufficiently small open covers. A section $s\in PU$ is mapped to the equivalence class it represents.


Hence the fact the above sheafification is zero when $f=1$ expresses the fact sheaf cohomology is global in nature (every cocycle admits a cover on which it's zero).


Using the above construction of sheafification, a section in $\mathbf Rf\_\ast F(V)$ is an equivalence class of a family of cocycles $(\Gamma\_i\in \mathrm H^\bullet(f^{-1}V\_i,F))$ where $(V\_i)\twoheadrightarrow V$ is an open cover, and we identify families if they coincide on a sufficiently small preimage of an open cover.


The sheafification map is non-zero for $n\geq 1$ if there's a cocycle of $F$ on some preimage $f^{-1}V$ which is not killed by any preimage of an open cover of $V$.


When $f:X\subset Y$ is a subspace inclusion the above means there's a cocycle of $F$ on $f^{-1}V=X\cap V$ such which does not restrict to zero on any open nighborhood in $f^{-1}V\_0=X\cap V\_0$ in $X$ of some problematic point $x\_0\in X$.


This can't happen for closed embeddings because their pushforward functor is exact.


**Question 1.** What's an instructive example of a subspace inclusion whose higher direct images are nonzero?


**Question 2.** For "what kind of maps" $f$ does one expect non-zero higher direct images non-zero? (Examples welcome.)


Lastly, I'd appreciate references with explicit topological examples the derived pushforward.



Subspace inclusion with non-vanishing higher direct images
You expect the higher direct images of $f: X \to Y$ to be nonzero if for arbitrarily small neighborhoods $y\_0 \in U$ the space $f^{-1}(U)$ has non-vanishing higher cohomology. I.e. $R^i f\_\* \mathbb Z$ vanishes if and only if all of its stalks do. 


For instance, think about the inclusion of $\mathbb R^2 - 0$ into $\mathbb R^2$. Taking $y\_0 = 0$, you see that if $U$ is a small ball around $y\_0$, then $f^{-1}(U) = U - 0$ is homotopy equivalent to a circle, which has $H^1 = \mathbb Z$


At every other point, the preimages of contractible neighborhoods are contractible. So, from looking at the stalks, we see that $R^1f\_\* \mathbb Z$ is $i\_\* \mathbb Z$ where $i$ is the inclusion of the origin. 


The takeaway is, to understand $R^i f\_\*$ you need to understand the cohomology of preimages $f^{-1}(U)$ for $U$ small.


