8 · Pair complementary degrees

Coherent duality pairs a class with one in the complementary degree. Their product reaches the top degree, and a trace map then produces a scalar on the base.

Try this. Rotate the surface and change its genus. Compare the number of classes on both sides, then follow a paired class through the trace.

Equal finite dimensions provide a classical model of a perfect pairing. Theorem 11.1 constructs compactly supported pushforward, the trace, and the duality isomorphism; this activity does not prove the theorem.