275:: an orientation of a differentiable manifold is an orientation of its tangent bundle. In particular, a differentiable manifold is orientable if and only if its tangent bundle is orientable as a vector bundle. (note: as a manifold, a tangent bundle is always orientable.)
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942:, Annals of Mathematics Studies, vol. 76, Princeton University Press; University of Tokyo Press,
248:. The multiplication (that is, cup product) by the Euler class of an oriented bundle gives rise to a
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359:, oriented bundles are classified by the infinite Grassmannian of oriented real vector spaces.
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From the cohomological point of view, for any ring Λ, a Λ-orientation of a real vector bundle
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866:. One can show, with some work, that the usual notion of an orientation coincides with a
663:{\displaystyle H^{*}(E;\Lambda )\to {\tilde {H}}^{*}(T(E);\Lambda ),x\mapsto x\smile u}
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796:{\displaystyle H^{*}(\pi ^{-1}(U);\Lambda )\to {\tilde {H}}^{*}(T(E|_{U});\Lambda )}
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181:), can be reduced to the subgroup consisting of those with positive determinant.
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892:) - this is used to formulate the Thom isomorphism for non-oriented bundles.
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and one demands that each trivialization map (which is a bundle map)
159:. In more concise terms, this says that the structure group of the
141:{\displaystyle \phi _{U}:\pi ^{-1}(U)\to U\times \mathbf {R} ^{n}}
355:
Just as a real vector bundle is classified by the real infinite
237:. A vector bundle that can be given an orientation is called an
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The notion of an orientation of a vector bundle generalizes an
859:{\displaystyle \pi ^{-1}(U)\simeq U\times \mathbf {R} ^{n}}
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A vector bundle together with an orientation is called an
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amounts to a reduction of the structure group to the
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329:{\displaystyle \operatorname {det} E=\wedge ^{n}E}
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27:Generalization of an orientation of a vector space
244:The basic invariant of an oriented bundle is the
512:{\displaystyle {\tilde {H}}^{*}(T(E);\Lambda )}
283:To give an orientation to a real vector bundle
63:, there is an orientation of the vector space
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151:is fiberwise orientation-preserving, where
381:means a choice (and existence) of a class
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908:Differential Forms in Algebraic Topology
291:is to give an orientation to the (real)
207:). In that situation, an orientation of
925:A Concise Course in Algebraic Topology.
340:. Similarly, to give an orientation to
42:; thus, given a real vector bundle π:
681:, that restricts to each isomorphism
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561:-module globally and locally: i.e.,
927:University of Chicago Press, 1999.
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554:{\displaystyle H^{*}(E;\Lambda )}
344:is to give an orientation to the
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264:is oriented in a canonical way.
188:is a real vector bundle of rank
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192:, then a choice of metric on
40:orientation of a vector space
211:amounts to a reduction from
881:integration along the fiber
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38:is a generalization of an
239:orientable vector bundle
221:special orthogonal group
677:), where "tilde" means
976:Orientation (geometry)
940:Characteristic classes
910:, New York: Springer,
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932:Milnor, John Willard
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169:general linear group
167:, which is the real
157:standard orientation
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50:, an orientation of
30:In mathematics, an
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886:Orientation bundle
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679:reduced cohomology
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346:unit sphere bundle
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971:Analytic geometry
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367:Main article:
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271:differentiable
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904:Bott, Raoul
246:Euler class
32:orientation
960:Categories
923:J.P. May,
897:References
519:as a free
459:generates
446:Thom space
369:Thom space
363:Thom space
279:Operations
34:of a real
842:×
836:≃
819:−
815:π
788:Λ
753:∗
746:~
736:→
730:Λ
710:−
706:π
697:∗
655:⌣
649:↦
637:Λ
614:∗
607:~
597:→
591:Λ
577:∗
546:Λ
532:∗
504:Λ
481:∗
474:~
426:Λ
395:∈
315:∧
305:
219:) to the
124:×
118:→
101:−
97:π
84:ϕ
938:(1974),
874:See also
377:of rank
287:of rank
273:manifold
256:Examples
946:
914:
944:ISBN
912:ISBN
888:(or
879:The
348:of
336:of
302:det
230:).
184:If
163:of
962::
934:;
352:.
260:A
252:.
241:.
224:SO
172:GL
868:Z
852:n
847:R
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830:U
827:(
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785:;
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777:U
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767:E
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