invented the cup and cap products and formulated Poincaré duality in these new terms.

Modern formulation

The modern statement of the Poincaré duality theorem is in terms of homology and cohomology: if M is a closed oriented n-manifold, then there is a canonically defined isomorphism for any integer k. To define such an isomorphism, one chooses a fixed fundamental class [M] of M, which will exist if is oriented.

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