TITLE: Why should an algebraic geometer care about singular / simplicial (co)homology?
QUESTION [22 upvotes]: I am a PhD student in algebraic / arithmetic geometry and I never took a formal course in algebraic topology, even though I have some basic knowledge.
In algebraic geometry we deal exclusively with sheaf cohomology since we care about non-constant sheaves. But I feel, maybe in my naivety, that a lot of the important results (for usual topological spaces) are only true for singular and simplicial cohomology when they coincide with sheaf cohomology (Alexander duality and "$H^i=0$ for $i>$ the covering dimension" come to mind).
With that in mind, I wonder if it is worth for someone with a similar background to study the details of a first course in algebraic topology. (Perhaps on the level of Hatcher's book.) Do I lose something by just thinking in terms of sheaf cohomology?

REPLY [8 votes]: I'm turning my comment above into an answer, as I realized there was more that I wanted.
IMO, if you're an arithmetic geometer the main motivation for learning singular cohomology is that it is intimately related to etale cohomology, in the sense that both of these are the simplest examples of a Weil cohomology theory.  I won't recall all the axioms but state the first two (at least as defined in the wikipedia article here):
Let $k$ be a field, $X/k$ a smooth projective variety of dimension $d$, and a $K$ a coefficient field of characteristic zero (for singular you often take $K = \mathbf{C}$ while for etale you take $K = \mathbf{Q}_\ell$)

$H^i(X)$ is finite dimensional.
$H^i(X)$ is zero for $i < 0$ and $i > 2d$.

In the topological setting, each of these is somewhat straightforward to prove because $X(\mathbf{C})$ has the structure of a real $2d$-dimensional CW-complex: One deduces (1) and (2) from the corresponding simplicial cochain complex that computes the cohomology of $X(\mathbf{C})$.
On the other hand in the etale setting these are not trivial to prove at all. For instance, the vanishing beyond degree $2d$ is a result of Artin for varieties over a separably closed field. To prove this result you need to fiber $X$ in curves to reduce to the case of a curve, and even then (may I say) the result is not trivial: Grothendieck was confused for a long time on how to compute the etale cohomology of curves!
As you keep going down the list of axioms for a Weil cohomology theory, you realize immediately that the corresponding statements in singular cohomology are much easier to prove. This alone should be enough to motivate you to care about singular cohomology.
Why is it important to realize that etale cohomology is a Weil cohomology theory? Well if you're an arithmetic geometer (with high probability) you will care about the Weil conjectures. The first two statements - rationality of zeta and the functional equation - pop out immediately as a consequence of the axioms of a cohomology theory. For instance, rationality is a consequence of the finite dimensionality of cohomology and the Lefschetz trace formula, while rationality from Poincare duality.