Chapter 0 of Griffith's Harris contains a substantial amount of topological results that I have not seen elsewhere, namely:
Poincare duality as a unimodular intersection pairing on homology. Also proved geometrically via a transversality argument and the dual cell decomposition.
The relation between Poincare duality and the intersection number of (n-k) and (k) dimensional cycles and sub-manifolds.
The fact that above form of Poincare duality with real coefficients 'is the same as' Poincare duality in the de Rham theory under the given isomorphisms.
Are there any other sources they treat these same ideas, perhaps with more detail? I have one further related question.
- Simplicial homology is easily shown to be isomorphic to the Cech theory with locally constant real valued functions. The de Rham isomorphism has a short sheaf theoretic proof relating the above Cech theory to the de Rham theory. Thus, we obtain by composition an isomorphism from the simplicial theory to the de Rham theory. Is there a source that unwraps this proof to show that the resulting isomorphism is induced by the usual integration pairing?