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Poincaré duality (Theorem)

If $ M$ is a compact, oriented, $ n$-dimensional manifold, then there is a canonical (though non-natural) isomorphism

$\displaystyle D:H^q(M,\mathbb{Z})\to H_{n-q}(M,\mathbb{Z})$
(where $ H^k(M,\mathbb{Z})$ is the $ k$th homology group of $ M$ with integer coefficients and $ H_k(M,\mathbb{Z})$ the $ k$th cohomology group) for all $ q$, which is given by cap product with a generator of $ H_n(M,\mathbb{Z})$ (a choice of a generator here corresponds to an orientation). This isomorphism exists with coefficients in $ \mathbb{Z}/2\mathbb{Z}$ regardless of orientation.

This isomorphism gives a nice interpretation to cup product. If $ X,Y$ are transverse submanifolds of $ M$, then $ X\cap Y$ is also a submanifold. All of these submanifolds represent homology classes of $ M$ in the appropriate dimensions, and

$\displaystyle D^{-1}([X])\cup D^{-1}([Y])=D^{-1}([X\cap Y]),$
where $ \cup$ is cup product, and $ \cap$ in intersection, not cap product.



"Poincaré duality" is owned by mathcam. [ full author list (3) | owner history (1) ]
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Other names:  Poincaré isomorphism
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Cross-references: intersection, dimensions, classes, homology, represent, submanifolds, transverse, cup product, interpretation, orientation, generator, cap product, group, coefficients, integer, homology group, isomorphism, canonical, manifold, oriented, compact
There are 4 references to this entry.

This is version 7 of Poincaré duality, born on 2002-12-04, modified 2004-11-30.
Object id is 3652, canonical name is PoincareDuality.
Accessed 3999 times total.

Classification:
AMS MSC: 55M05 (Algebraic topology :: Classical topics :: Duality)
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