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| In [[mathematics]], '''local coefficients''' is an idea from [[algebraic topology]], a kind of half-way stage between [[homology theory]] or [[cohomology theory]] with coefficients in the usual sense, in a fixed [[abelian group]] ''A'', and general [[sheaf cohomology]] which, roughly speaking, allows coefficients to vary from point to point in a [[topological space]] ''X''. Such a concept was introduced by [[Norman Steenrod]].
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| ==Formal definition==
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| Let ''X'' be a [[locally path connected]] topological space, and ''M'' a module over some ring ''R''. A local coefficient system of ''R''-modules ''E'' with fiber ''M'' is a locally trivial fibration (i.e. a [[fiber bundle]]) with fiber ''M'' with an action of the [[fundamental groupoid]] of the base ''X'', that is, for each path <math>\gamma : [0,1]\to X</math>, a morphism <math>\gamma_*: E_{\gamma(0)}\to E_{\gamma(1)}</math> that depends only on the homotopy class with fixed extremities of the path, is the identity on constant paths and such that composition of paths corresponds to compositions of morphisms.
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| In [[sheaf theory]] terms, a [[constant sheaf]] has [[locally constant function]]s as its sections. Consider instead a sheaf ''F'', such that locally on ''X'' it is a constant sheaf. That means that in some neighbourhood of any ''x'' in ''X'', it is isomorphic to a constant sheaf. Then ''F'' may be used as a system of ''local coefficients'' on ''X''.
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| ==Applications==
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| Examples arise geometrically from [[vector bundle]]s with [[flat connection]]s, and from topology by means of [[linear representation]]s of the [[fundamental group]].
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| The cohomology with local coefficients in the module corresponding to the [[orientation covering]] can be used to formulate [[Poincaré duality]] for non-orientable manifolds: see [[Twisted Poincaré duality]].
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| Larger classes of sheaves are useful: for example the idea of a [[constructible sheaf]] in [[algebraic geometry]]. These turn out, approximately, to be local coefficients away from a singular set.
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| ==External links==
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| *[http://math.stackexchange.com/questions/13332/what-local-system-really-is A discussion of the notion] on [[Stack exchange]]
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| {{DEFAULTSORT:Local System}}
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| [[Category:Sheaf theory]]
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| [[Category:Algebraic topology]]
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Hello and welcome. My title is Figures Wunder. My day occupation is a meter reader. One of the extremely best things in the world for me is to do aerobics and now I'm attempting to earn money with it. For many years he's been residing in North Dakota and his family enjoys it.
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