Editing Orientability (section)

===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>