Orientation of a vector bundle

<h2 id="examples">Examples</h2>
A <a href="/facts/Complex_vector_bundle/0nE8TEIn">complex vector bundle</a> is oriented in a canonical way.
The notion of an orientation of a vector bundle generalizes an <a href="/facts/Orientation_of_a_manifold/1SEQM2Pt">orientation of a differentiable manifold</a>: 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.)

<h2 id="operations">Operations</h2>
To give an orientation to a real vector bundle E of rank n is to give an orientation to the (real) <a href="/facts/Determinant_bundle/CupAl1ft">determinant bundle</a> 
 
 
 
 det
 ⁡
 E
 =
 
 ∧
 
 n
 
 
 E
 
 
 {\displaystyle \operatorname {det} E=\wedge ^{n}E}
 
 of E. Similarly, to give an orientation to E is to give an orientation to the <a href="/facts/Unit_sphere_bundle/Qf2Yybh5">unit sphere bundle</a> of E.
Just as a real vector bundle is classified by the real infinite <a href="/facts/Grassmannian/QcmgLRq8">Grassmannian</a>, oriented bundles are classified by the infinite Grassmannian of oriented real vector spaces.

<h2 id="thom-space">Thom space</h2>
Main article: <a href="/facts/Thom_space/5tgFlV5m">Thom space</a>
From the cohomological point of view, for any ring Λ, a Λ-orientation of a real vector bundle E of rank n means a choice (and existence) of a class

u
        ∈
        
          H
          
            n
          
        
        (
        T
        (
        E
        )
        ;
        Λ
        )
      
    
    {\displaystyle u\in H^{n}(T(E);\Lambda )}

in the cohomology ring of the <a href="/facts/Thom_space/5tgFlV5m">Thom space</a> T(E) such that u generates 
 
 
 
 
 
 
 
 H
 ~
 
 
 
 
 ∗
 
 
 (
 T
 (
 E
 )
 ;
 Λ
 )
 
 
 {\displaystyle {\tilde {H}}^{*}(T(E);\Lambda )}
 
 as a free 
 
 
 
 
 H
 
 ∗
 
 
 (
 E
 ;
 Λ
 )
 
 
 {\displaystyle H^{*}(E;\Lambda )}
 
-module globally and locally: i.e.,

H
          
            ∗
          
        
        (
        E
        ;
        Λ
        )
        →
        
          
            
              
                H
                ~
              
            
          
          
            ∗
          
        
        (
        T
        (
        E
        )
        ;
        Λ
        )
        ,
        x
        ↦
        x
        ⌣
        u
      
    
    {\displaystyle H^{*}(E;\Lambda )\to {\tilde {H}}^{*}(T(E);\Lambda ),x\mapsto x\smile u}

is an isomorphism (called the <a href="/facts/Thom_isomorphism/5tgFlV5m">Thom isomorphism</a>), where "tilde" means <a href="/facts/Reduced_cohomology/ySoPPB63">reduced cohomology</a>, that restricts to each isomorphism

H
          
            ∗
          
        
        (
        
          π
          
            −
            1
          
        
        (
        U
        )
        ;
        Λ
        )
        →
        
          
            
              
                H
                ~
              
            
          
          
            ∗
          
        
        (
        T
        (
        E
        
          
            |
          
          
            U
          
        
        )
        ;
        Λ
        )
      
    
    {\displaystyle H^{*}(\pi ^{-1}(U);\Lambda )\to {\tilde {H}}^{*}(T(E|_{U});\Lambda )}

induced by the trivialization 
 
 
 
 
 π
 
 −
 1
 
 
 (
 U
 )
 ≃
 U
 ×
 
 
 R
 
 
 n
 
 
 
 
 {\displaystyle \pi ^{-1}(U)\simeq U\times \mathbf {R} ^{n}}
 
. One can show, with some work, that the usual notion of an orientation coincides with a Z-orientation.

<h2 id="see-also">See also</h2>
<ul><li>The <a href="/facts/Integration_along_the_fiber/BPpfJ8HD">integration along the fiber</a></li>
<li><a href="/facts/Orientation_bundle/cThhN158">Orientation bundle</a> (or <a href="/facts/Orientation_sheaf/cThhN158">orientation sheaf</a>) - this is used to formulate the Thom isomorphism for non-oriented bundles.</li></ul>

<ul><li><a href="/facts/Raoul_Bott/5JHkhPx8">Bott, Raoul</a>; Tu, Loring (1982), Differential Forms in Algebraic Topology, New York: Springer, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-387-90613-4</li>
<li>J.P. May, A Concise Course in Algebraic Topology. University of Chicago Press, 1999.</li>
<li><a href="/facts/John_Milnor/LTf24paN">Milnor, John Willard</a>; <a href="/facts/Jim_Stasheff/kceG3XPX">Stasheff, James D.</a> (1974), Characteristic classes, Annals of Mathematics Studies, vol. 76, Princeton University Press; University of Tokyo Press, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-691-08122-9</li></ul>

Orientation of a vector bundle open-in-new

Orientation of a vector bundle