Cap product with the locally finite fundamental class of an -manifold without boundary gives the displayed duality, including a possibly nontrivial orientation local system . It is the noncompact version of Poincare duality; for a compact manifold Borel-Moore homology is ordinary homology. Locally it is the relative fundamental-class calculation in an oriented coordinate ball; compatibility on overlaps and the Mayer–Vietoris sequence give the global statement. Since the chain complex of locally finite homology is zero in negative degrees, this duality implies for .
For a continuous map of nonzero mapping degree between closed connected oriented manifolds of the same dimension, is injective in every degree. Pair a nonzero class with a complementary class using Poincare duality. Naturality gives , so . Over coefficients of positive characteristic the argument requires that the degree remain nonzero in that field.
For a connected closed oriented four-manifold, Poincare duality identifies with . The universal coefficient theorem for cohomology identifies with because has zero Ext. Consequently is torsion-free and its rank equals that of .
For a closed connected manifold, cap product with gives the displayed isomorphism. The ordinary integral theorem results when the orientation system is trivial. Modulo two the sign representation is always trivial, so duality holds without an orientability assumption. In particular a nonorientable connected closed -manifold has and .
Zero Euler characteristic and give free rank . The universal coefficient theorem for cohomology identifies with , so the only torsion in is one copy of . In particular .
Represent one-dimensional homology classes on a closed oriented topological surface by transverse oriented curves and sum their signed crossing points. This gives a bilinear form that is skew-symmetric. A symplectic handle basis has matrix , which is unimodular. In particular, a class detected with intersection or by another integral class is a primitive homology class: the detecting homomorphism splits its rank-one subgroup.
For a closed -dimensional manifold, Poincare duality holds over without an orientation assumption. The mod-two fundamental class exists because orientation signs disappear. The resulting complementary-degree intersection pairing is a perfect pairing.
For a closed oriented manifold of dimension , Poincare duality pairs all rational Betti numbers except the middle one, making their total contribution to the Euler characteristic even. The middle cohomology degree is odd, so its nondegenerate Poincare duality pairing is an alternating bilinear form and has even dimension. Hence the Euler characteristic is even. Nonorientable manifolds can have odd Euler characteristic, as Real projective space of even dimension demonstrates.
Take a connected sum of copies of Complex projective space and copies of , with the empty sum interpreted as . Their Euler characteristics are four, zero and two, respectively. Removing a six-ball subtracts one from the Euler characteristic and gluing along its boundary subtracts zero, so each connected sum subtracts two. The result is . For any even integer , choose and .
For a map between closed oriented manifolds of the same dimension, Poincare duality turns homological pushforward into a degree-preserving map on cohomology. It is adjoint to pullback under the duality pairing: . If has degree , nondegeneracy gives . Maps of different dimensions require the corresponding degree shift.
For a closed oriented -dimensional manifold over a field , the cup product pairingis a perfect pairing between complementary degrees. The cap product with the fundamental class is an isomorphism by Poincare duality, and the universal coefficient theorem for cohomology over identifies the complementary cohomology with the full dual space of the resulting homology. Evaluation is therefore a perfect pairing. Taking the top-degree part of the same formula defines a nondegenerate bilinear form on the full graded cohomology. A boundary requires relative groups and Poincare-Lefschetz duality instead.
For a closed oriented manifold, the rational middle-dimensional cup-product pairing is nondegenerate by Poincare duality. Its degree is odd, so graded commutativity makes it skew-symmetric. A nondegenerate skew-symmetric form has even dimension. In particular a closed oriented six-manifold cannot have a rank-one third cohomology group.
// Target: algebra.bigb
If is a map between closed connected oriented manifolds of the same dimension and its degree is nonzero in a coefficient field , then is injective. For a nonzero class , Poincare duality supplies with , and naturality gives
For a compact oriented -manifold with boundary, cap product with its relative fundamental class gives
Reverse a handle decomposition of a compact -dimensional manifold. Absolute -handles become relative -handles based on the boundary. The absolute boundary matrices count attaching sphere-belt sphere intersections; the corresponding relative matrices are their transposes over . The absolute chain complex is thus the complementary-degree dual of the relative chain complex. The universal coefficient theorem for cohomology proves Poincare-Lefschetz duality without an orientation hypothesis.
For an oriented codimension- closed submanifold , its Poincare dual is the class in whose cup-product evaluation against any complementary class equals intersection with .
For a compact oriented submanifold without boundary in an oriented manifold, the normal Thom class gives a compactly supported cohomology class dual to its pushed-forward fundamental class. It is independent of the tubular neighborhood and its ordinary-cohomology image vanishes on the complement of the submanifold. A noncompact submanifold closed only as a subset generally has its dual in ordinary cohomology, using locally finite homology instead.
If a closed oriented manifold is covered by two open sets and a positive-degree cohomology class restricts to zero on both, then its cap product with the fundamental class vanishes in every degree strictly below the top. A chain proof uses the small simplex theorem and replaces the two restricted cocycles by coboundaries.
A degree-one map between closed connected oriented manifolds of the same dimension induces an injection . The wrong-way map defined using Poincare duality is a left inverse by the projection formula and .
An integral homology -sphere is a space with the same integral homology as . Removing one point from a closed oriented integral homology sphere produces an acyclic space by the long exact sequence of the pair and the local orientation isomorphism in degree .
A simply connected closed three-dimensional manifold is orientable, since its orientation monodromy factors through its trivial fundamental group. Its first integral homology is the abelianization of that fundamental group and is zero. The universal coefficient theorem for cohomology gives , and Poincare duality gives , . Connectedness gives , and duality gives zero homology above dimension three. Thus it is an integral homology sphere, without needing any homeomorphism classification.
A homotopy sphere is a closed manifold with the homotopy type of a sphere of the same dimension. In dimensions at least two, a closed simply connected integral homology sphere is a homotopy sphere: successive applications of the Hurewicz theorem produce a map from the sphere inducing an isomorphism on integral homology, and the homological Whitehead theorem makes it a homotopy equivalence. The topological generalized Poincare theorem identifies its topological type, while prescribed smooth structures require a separate question.
Every closed topological manifold with the homotopy type of is homeomorphic to . This is a classification theorem about homeomorphisms, and does not assert a diffeomorphism between prescribed smooth structures. It is the precise final step needed after proving that a cone link is a simply connected integral homology sphere in the manifold criterion for a suspension.
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Poincaré duality is a fundamental theorem in algebraic topology that describes a duality relationship between certain topological spaces, particularly manifolds, and their cohomology groups. Named after the French mathematician Henri Poincaré, the theorem specifically applies to compact, oriented manifolds.