In mathematics, the twisted Poincaré duality is a theorem removing the restriction on Poincaré duality to oriented manifolds. The existence of a global orientation is replaced by carrying along local information, by means of a

local coefficient system In mathematics, a local system (or a system of local coefficients) on a topological space ''X'' is a tool from algebraic topology which interpolates between cohomology with coefficients in a fixed abelian group ''A'', and general sheaf cohomology ...

Twisted Poincaré duality for de Rham cohomology

Another version of the theorem with real coefficients features

de Rham cohomology In mathematics, de Rham cohomology (named after Georges de Rham) is a tool belonging both to algebraic topology and to differential topology, capable of expressing basic topological information about smooth manifolds in a form particularly adapte ...

with values in the orientation bundle. This is the flat real

line bundle In mathematics, a line bundle expresses the concept of a line that varies from point to point of a space. For example, a curve in the plane having a tangent line at each point determines a varying line: the ''tangent bundle'' is a way of organisin ...

denoted

o(M)

, that is trivialized by coordinate charts of the manifold

M

, with transition maps the sign of the Jacobian determinant 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 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. 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 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 Poincaré duality 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
the answers to this thread
on MathOverflow. *The online boo
''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