In mathematics, the twisted Poincaré duality is a theorem removing the restriction on

Poincaré duality In mathematics, the Poincaré duality theorem, named after Henri Poincaré, is a basic result on the structure of the homology and cohomology groups of manifolds. It states that if ''M'' is an ''n''-dimensional oriented closed manifold (compact ...

oriented manifold In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "counterclockwise". A space i ...

s. The existence of a global orientation is replaced by carrying along local information, by means of a local coefficient system.

Twisted Poincaré duality for de Rham cohomology

Another version of the theorem with real coefficients features de Rham cohomology with values in the orientation bundle. This is the

flat Flat or flats may refer to: Architecture * Flat (housing), an apartment in the United Kingdom, Ireland, Australia and other Commonwealth countries Arts and entertainment * Flat (music), a symbol () which denotes a lower pitch * Flat (soldier), ...

real line bundle denoted

o(M)

, that is trivialized by coordinate charts of the manifold

M

, with transition maps the sign of the

Jacobian determinant In vector calculus, the Jacobian matrix (, ) of a vector-valued function of several variables is the matrix of all its first-order partial derivatives. When this matrix is square, that is, when the function takes the same number of variables ...

of the charts transition maps. As a flat line bundle, it has a de Rham cohomology, denoted by :

H^* (M; \R^w)

H^* (M; o(M))

. For ''M'' a

compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in British ...

manifold, the top degree cohomology is equipped with a so-called trace morphism :

\theta\colon H^d (M; o(M)) \to \R

, that is to be interpreted as integration on ''M'', ''i.e.'', evaluating against the

fundamental class In mathematics, the fundamental class is a homology class 'M''associated to a connected orientable compact manifold of dimension ''n'', which corresponds to the generator of the homology group H_n(M,\partial M;\mathbf)\cong\mathbf . The fundam ...

. Poincaré duality for differential forms is then the conjunction, for ''M'' connected, of the following two statements: * The trace morphism is a linear isomorphism. * The cup product, or

exterior product In mathematics, specifically in topology, the interior of a subset of a topological space is the union of all subsets of that are open in . A point that is in the interior of is an interior point of . The interior of is the complement of th ...

of differential forms :

\cup \colon H^* (M; \R)\otimes H^(M, o(M)) \to H^d(M, o(M)) \simeq \R

is non-degenerate. The oriented

is contained in this statement, as understood from the fact that the orientation bundle ''o(M)'' is trivial if the manifold is oriented, an orientation being a global trivialization, ''i.e.'', a nowhere vanishing parallel section.

References

*Some references are provided i
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''Algebraic and geometric surgery''
by Andrew Ranicki. * {{DEFAULTSORT:Twisted Poincare duality Algebraic topology Manifolds Duality theories Theorems in topology

Twisted Poincaré duality for de Rham cohomology

See also

References