2 qualify "poincare's proof"; added 31 characters in body

I'd say the standard definition of singular homology is pretty bad.

It's a historical relic in some sense -- topologists were so concerned by naturality, whether manifolds have combinatorially distinct triangulations and issues such as that, that they decided those preoccupations were more important than imparting a solid foundational intuition as to what a homology class is.

In my experience, people who see Poincare's proof of Poincare duality first vs. the people who see a singular homology exposition usually have a far better command of what is actually going on, to the point where they view Poincare duality is something light and natural, while most students that see it through the eyes of singular homology more often see it as something distant and intractible.

And all that effort is for what? So students can know Poincare duality is true on topological manifolds, when all the examples they've seen are smooth manifolds.

edit: my preferred way to describe Poincare's proof is to modernize it a tad. Your set-up is a triangulated manifold $M$, then you construct the dual polyhedral decomposition (a CW-decomposition) so that the (simplicial) $i$-cells of $M$ are in bijective correspondence with the (dual polyhedral) $m-i$-cells of $M$. This is much more straightforward than living in the simplicial world. Then you show that (up to a sign change) the chain complex for the simplicial homology is the chain complex for the cohomology of the dual polyhedral decomposition. The fussiest bit is keeping track of the orientations in the orientable case.