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Poincaré Duality

Poincaré duality states that for a connected closed orientable manifold M of dimension n, a field F, and the fundamental class [M] in H_n(M;F), taking the cap product with [M] gives an isomorphism

H^k(M;F)->H_(n-k)(M;F), alpha|->[M]⌢alpha.

Consequently, the Betti numbers satisfy

b_k(M)=b_(n-k)(M).

Under this correspondence, the cup product of cohomology classes is dual to the intersection product of homology classes. For a compact orientable manifold with boundary, the corresponding statement is the Poincaré-Lefschetz duality isomorphism H^k(M;F)=H_(n-k)(M,partialM;F). With coefficients in F_2, an orientation is not required.

See also

Betti Number, Cohomology, Cup Product, Fundamental Class, Homology, Homology Intersection

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References

Hatcher, A. Algebraic Topology. Cambridge, England: Cambridge University Press, 2002. https://pi.math.cornell.edu/~hatcher/AT/AT.pdf.Munkres, J. R. Elements of Algebraic Topology. Menlo Park, CA: Addison-Wesley, 1984.

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Poincaré Duality

Cite this as:

Weisstein, Eric W. "Poincaré Duality." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/PoincareDuality.html

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