Editing Orientability (section)

==Orientability of manifolds==

Let ''M'' be a connected topological ''n''-[[manifold (mathematics)|manifold]].  There are several possible definitions of what it means for ''M'' to be orientable.  Some of these definitions require that ''M'' has extra structure, like being differentiable.  Occasionally, {{math|1=''n'' = 0}} must be made into a special case.  When more than one of these definitions applies to ''M'', then ''M'' is orientable under one definition if and only if it is orientable under the others.<ref>{{Cite book | last1=Spivak | first1=Michael | author1-link=Michael Spivak | title=Calculus on Manifolds | publisher=[[HarperCollins]] | isbn=978-0-8053-9021-6 | year=1965}}</ref><ref>{{Cite book | last1=Hatcher | first1=Allen | author1-link=Allen Hatcher | title=Algebraic Topology | publisher=[[Cambridge University Press]] | isbn=978-0521795401 | year=2001}}</ref>

===Orientability of differentiable manifolds===

The most intuitive definitions require that ''M'' be a differentiable manifold.  This means that the transition functions in the atlas of ''M'' are ''C''<sup>1</sup>-functions.  Such a function admits a [[Jacobian determinant]].  When the Jacobian determinant is positive, the transition function is said to be '''orientation preserving'''.  An '''oriented atlas''' on ''M'' is an atlas for which all transition functions are orientation preserving.  ''M'' is '''orientable''' if it admits an oriented atlas.  When {{math|''n'' &gt; 0}}, an '''orientation''' of ''M'' is a maximal oriented atlas.  (When {{math|1=''n'' = 0}}, an orientation of ''M'' is a function {{math|''M'' → {±1}<nowiki/>}}.)

Orientability and orientations can also be expressed in terms of the tangent bundle.  The tangent bundle is a [[vector bundle]], so it is a [[fiber bundle]] with [[structure group]] {{math|GL(''n'', '''R''')}}.  That is, the transition functions of the manifold induce transition functions on the tangent bundle which are fiberwise linear transformations.  If the structure group can be reduced to the group {{math|GL<sup>+</sup>(''n'', '''R''')}} of positive determinant matrices, or equivalently if there exists an atlas whose transition functions determine an orientation preserving linear transformation on each tangent space, then the manifold ''M'' is orientable.  Conversely, ''M'' is orientable if and only if the structure group of the tangent bundle can be reduced in this way.  Similar observations can be made for the frame bundle.

Another way to define orientations on a differentiable manifold is through [[volume form]]s.  A volume form is a nowhere vanishing section ''&omega;'' of {{math|⋀{{sup|''n''}} ''T''{{i sup|∗}}''M''}}, the top exterior power of the cotangent bundle of ''M''.  For example, '''R'''<sup>''n''</sup> has a standard volume form given by {{math|''dx''<sup>1</sup> ∧ ⋯ ∧ ''dx''<sup>''n''</sup>}}.  Given a volume form on ''M'', the collection of all charts {{math|''U'' → '''R'''<sup>''n''</sup>}} for which the standard volume form pulls back to a positive multiple of ''&omega;'' is an oriented atlas.  The existence of a volume form is therefore equivalent to orientability of the manifold.

Volume forms and tangent vectors can be combined to give yet another description of orientability.  If {{math|''X''<sub>1</sub>, …, ''X''<sub>''n''</sub>}} is a basis of tangent vectors at a point ''p'', then the basis is said to be '''right-handed''' if {{math|&omega;(''X''<sub>1</sub>, …, ''X''<sub>''n''</sub>) &gt; 0}}.  A transition function is orientation preserving if and only if it sends right-handed bases to right-handed bases.  The existence of a volume form implies a reduction of the structure group of the tangent bundle or the frame bundle to {{math|GL<sup>+</sup>(''n'', '''R''')}}.  As before, this implies the orientability of ''M''.  Conversely, if ''M'' is orientable, then local volume forms can be patched together to create a global volume form, orientability being necessary to ensure that the global form is nowhere vanishing.

===Homology and the orientability of general manifolds===

At the heart of all the above definitions of orientability of a differentiable manifold is the notion of an orientation preserving transition function.  This raises the question of what exactly such transition functions are preserving.  They cannot be preserving an orientation of the manifold because an orientation of the manifold is an atlas, and it makes no sense to say that a transition function preserves or does not preserve an atlas of which it is a member.

This question can be resolved by defining local orientations.  On a one-dimensional manifold, a local orientation around a point ''p'' corresponds to a choice of left and right near that point.  On a two-dimensional manifold, it corresponds to a choice of clockwise and counter-clockwise.  These two situations share the common feature that they are described in terms of top-dimensional behavior near ''p'' but not at ''p''.  For the general case, let ''M'' be a topological ''n''-manifold.  A '''local orientation''' of ''M'' around a point ''p'' is a choice of generator of the group
:<math>H_n\left(M, M \setminus \{p\}; \mathbf{Z}\right).</math>

To see the geometric significance of this group, choose a chart around ''p''.  In that chart there is a neighborhood of ''p'' which is an open ball ''B'' around the origin ''O''.  By the [[excision theorem]], <math>H_n\left(M, M \setminus \{p\}; \mathbf{Z}\right)</math> is isomorphic to <math>H_n\left(B, B \setminus \{O\}; \mathbf{Z}\right)</math>.  The ball ''B'' is contractible, so its homology groups vanish except in degree zero, and the space {{math|''B'' \ ''O''}} is an {{math|(''n'' − 1)}}-sphere, so its homology groups vanish except in degrees {{math|''n'' − 1}} and {{math|0}}.  A computation with the [[long exact sequence]] in [[relative homology]] shows that the above homology group is isomorphic to <math>H_{n-1}\left(S^{n-1}; \mathbf{Z}\right) \cong \mathbf{Z}</math>.  A choice of generator therefore corresponds to a decision of whether, in the given chart, a sphere around ''p'' is positive or negative.  A reflection of {{math|'''R'''<sup>''n''</sup>}} through the origin acts by negation on <math>H_{n-1}\left(S^{n-1}; \mathbf{Z}\right)</math>, so the geometric significance of the choice of generator is that it distinguishes charts from their reflections.

On a topological manifold, a transition function is '''orientation preserving''' if, at each point ''p'' in its domain, it fixes the generators of <math>H_n\left(M, M \setminus \{p\}; \mathbf{Z}\right)</math>.  From here, the relevant definitions are the same as in the differentiable case.  An '''oriented atlas''' is one for which all transition functions are orientation preserving, ''M'' is '''orientable''' if it admits an oriented atlas, and when {{math|''n'' &gt; 0}}, an '''orientation''' of ''M'' is a maximal oriented atlas.

Intuitively, an orientation of ''M'' ought to define a unique local orientation of ''M'' at each point.  This is made precise by noting that any chart in the oriented atlas around ''p'' can be used to determine a sphere around ''p'', and this sphere determines a generator of <math>H_n\left(M, M \setminus \{p\}; \mathbf{Z}\right)</math>.  Moreover, any other chart around ''p'' is related to the first chart by an orientation preserving transition function, and this implies that the two charts yield the same generator, whence the generator is unique.

Purely homological definitions are also possible.  Assuming that ''M'' is closed and connected, ''M'' is '''orientable''' if and only if the ''n''th homology group <math>H_n(M; \mathbf{Z})</math> is isomorphic to the integers '''Z'''.  An '''orientation''' of ''M'' is a choice of generator {{math|&alpha;}} of this group.  This generator determines an oriented atlas by fixing a generator of the infinite cyclic group <math>H_n(M ; \mathbf{Z})</math> and taking the oriented charts to be those for which {{math|&alpha;}} pushes forward to the fixed generator. Conversely, an oriented atlas determines such a generator as compatible local orientations can be glued together to give a generator for the homology group <math>H_n(M ; \mathbf{Z})</math>.<ref>{{harvnb|Hatcher|2001|p=236 Theorem 3.26(a)}}</ref>

===Orientation and cohomology===

A manifold ''M'' is orientable if and only if the first [[Stiefel–Whitney class]] <math>w_1(M) \in H^1(M; \mathbf{Z}/2)</math> vanishes. In particular, if the first cohomology group with '''Z'''/2 coefficients is zero, then the manifold is orientable. Moreover, if ''M'' is orientable and ''w''<sub>1</sub> vanishes, then <math>H^0(M; \mathbf{Z}/2)</math> parametrizes the choices of orientations.<ref>{{Cite book | last1=Lawson | first1=H. Blaine | author1-link=H. Blaine Lawson |  last2=Michelsohn | first2=Marie-Louise | author2-link=Marie-Louise Michelsohn | title=Spin Geometry | publisher=[[Princeton University Press]] | isbn=0-691-08542-0 | year=1989 |page=79 Theorem 1.2}}</ref> This characterization of orientability extends to [[orientation of a vector bundle|orientability of general vector bundles]] over ''M'', not just the tangent bundle.

===The orientation double cover===

Around each point of ''M'' there are two local orientations.  Intuitively, there is a way to move from a local orientation at a point {{math|''p''}} to a local orientation at a nearby point {{math|''p''&prime;}}: when the two points lie in the same coordinate chart {{math|''U'' → '''R'''<sup>''n''</sup>}}, that coordinate chart defines compatible local orientations at {{math|''p''}} and {{math|''p''&prime;}}.  The set of local orientations can therefore be given a topology, and this topology makes it into a manifold.

More precisely, let ''O'' be the set of all local orientations of ''M''.  To topologize ''O'' we will specify a subbase for its topology.  Let ''U'' be an open subset of ''M'' chosen such that <math>H_n(M, M \setminus U; \mathbf{Z})</math> is isomorphic to '''Z'''.  Assume that &alpha; is a generator of this group.  For each ''p'' in ''U'', there is a pushforward function <math>H_n(M, M \setminus U; \mathbf{Z}) \to H_n\left(M, M \setminus \{p\}; \mathbf{Z}\right)</math>.  The codomain of this group has two generators, and &alpha; maps to one of them.  The topology on ''O'' is defined so that
:<math>\{\text{Image of } \alpha \text{ in } H_n\left(M, M \setminus \{p\}; \mathbf{Z}\right) \colon p \in U\}</math>
is open.

There is a canonical map {{math|&pi; : ''O'' → ''M''}} that sends a local orientation at ''p'' to ''p''.  It is clear that every point of ''M'' has precisely two preimages under {{math|&pi;}}.  In fact, {{math|&pi;}} is even a local homeomorphism, because the preimages of the open sets ''U'' mentioned above are homeomorphic to the disjoint union of two copies of ''U''.  If ''M'' is orientable, then ''M'' itself is one of these open sets, so ''O'' is the disjoint union of two copies of ''M''.  If ''M'' is non-orientable, however, then ''O'' is connected and orientable.  The manifold ''O'' is called the '''orientation double cover'''.

===Manifolds with boundary===

If ''M'' is a manifold with boundary, then an orientation of ''M'' is defined to be an orientation of its interior.  Such an orientation induces an orientation of ∂''M''.  Indeed, suppose that an orientation of ''M'' is fixed.  Let {{math|''U'' → '''R'''<sup>''n''</sup><sub>+</sub>}} be a chart at a boundary point of ''M'' which, when restricted to the interior of ''M'', is in the chosen oriented atlas.  The restriction of this chart to ∂''M'' is a chart of ∂''M''.  Such charts form an oriented atlas for ∂''M''.

When ''M'' is smooth, at each point ''p'' of ∂''M'', the restriction of the tangent bundle of ''M'' to ∂''M'' is isomorphic to {{math|''T''<sub>''p''</sub>∂''M'' ⊕ '''R'''}}, where the factor of '''R''' is described by the inward pointing normal vector.  The orientation of ''T''<sub>''p''</sub>∂''M'' is defined by the condition that a basis of ''T''<sub>''p''</sub>∂''M'' is positively oriented if and only if it, when combined with the inward pointing normal vector, defines a positively oriented basis of ''T''<sub>''p''</sub>''M''.