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Finite-type (Vassiliev) invariants, quantum invariants, and perturbative invariants of knotted objects and of manifolds.
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Quantum homology of $(S^2 \times S^2,\omega_{FS}\oplus \omega_{FS})$ and Poincare duality
A bit late for this one, but I'll still post the answer for future visitors.
Poincaré duality on the quantum homology is just the same as Poincaré duality on normal homology, see for example the famou …