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| | She is known by the name of Myrtle Shryock. For many years I've been operating as a payroll clerk. Minnesota is exactly where he's been living for years. Body developing is what my family members and I enjoy.<br><br>Also visit my web page; [http://youulike.com/profile/116412 at home std testing] |
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| In [[algebraic topology]], a branch of [[mathematics]], an ''orientation character'' on a [[group (mathematics)|group]] <math>\pi</math> is a [[group homomorphism]]
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| :<math>\omega\colon \pi \to \left\{\pm 1\right\}</math>. This notion is of particular significance in [[surgery theory]].
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| ==Motivation==
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| Given a [[manifold]] ''M'', one takes <math>\pi=\pi_1 M</math> (the [[fundamental group]]), and then <math>\omega</math> sends an element of <math>\pi</math> to <math>-1</math> if and only if the class it represents is orientation-reversing.
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| This map <math>\omega</math> is trivial if and only if ''M'' is [[orientable]].
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| The orientation character is an algebraic structure on the fundamental group of a manifold, which captures which loops are orientation reversing and which are orientation preserving.
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| ==Twisted group algebra==
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| The orientation character defines a twisted involution ([[*-ring]] structure) on the [[group ring]] <math>\mathbf{Z}[\pi]</math>, by <math>g \mapsto \omega(g)g^{-1}</math> (i.e., <math>\pm g^{-1}</math>, accordingly as <math>g</math> is orientation preserving or reversing). This is denoted <math>\mathbf{Z}[\pi]^\omega</math>.
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| ==Examples==
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| *In [[real projective space]]s, the orientation character evaluates trivially on loops if the dimension is odd, and assigns -1 to noncontractible loops in even dimension.
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| ==Properties==
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| The orientation character is either trivial or has kernel an index 2 subgroup, which determines the map completely.
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| ==See also==
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| * [[Whitney characteristic class]]
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| * [[Local system]]
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| * [[Twisted Poincaré duality]]
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| == External links ==
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| *[http://www.map.mpim-bonn.mpg.de/Orientation_character Orientation character] at the Manifold Atlas
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| [[Category:Geometric topology]]
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She is known by the name of Myrtle Shryock. For many years I've been operating as a payroll clerk. Minnesota is exactly where he's been living for years. Body developing is what my family members and I enjoy.
Also visit my web page; at home std testing