Orientability: Difference between revisions

In [[mathematics]], '''orientability''' is a property of some [[topological space]]s such as [[real vector space]]s, [[Euclidean space]]s, [[Surface (topology)|surface]]s, and more generally [[manifold]]s that allows a consistent definition of "clockwise" and "counterclockwise".<ref>{{Cite book|url=https://books.google.com/books?id=9MQ-AAAAIAAJ&q=oriented+manifold|title=Modern multidimensional calculus|last=Munroe|first=Marshall Evans|date=1963|publisher=Addison-Wesley Pub. Co.|language=en|page=263}}</ref> A space is '''orientable''' if such a consistent definition exists. In this case, there are two possible definitions, and a choice between them is an '''orientation''' of the space. Real vector spaces, Euclidean spaces, and [[sphere]]s are orientable. A space is '''non-orientable''' if "clockwise" is changed into "counterclockwise" after running through some [[loop (topology)|loop]]s in it, and coming back to the starting point. This means that a [[geometric shape]], such as [[File:Small pie.svg|20px]], that moves continuously along such a loop is changed into its own [[mirror image]] [[File:pie 2.svg|20px]]. A [[Möbius strip]] is an example of a non-orientable space.

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>{{Cite book harvnb| last1=Hatcher | first1=Allen 2001| author1-linkp=Allen Hatcher | title=Algebraic Topology | publisher=[[Cambridge University Press]] | isbn=978-0521795401 | year=2001}},236 Theorem 3.26(a) on p. 236}}</ref>

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''₁ 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 on p. 79}}</ref> This characterization of orientability extends to [[orientation of a vector bundle|orientability of general vector bundles]] over ''M'', not just the tangent bundle.

In [[Lorentzian geometry]], there are two kinds of orientability: [[space orientability]] and [[time orientability]]. These play a role in the [[causal structure]] of spacetime.<ref>{{cite book | authorauthor1-link=[[Stephen Hawking |first1=S.W. |last1=Hawking]], [[|author2-link=George Francis Rayner Ellis |first2=G.F.R. |last2=Ellis]] | title=The Large Scale Structure of Space-Time | location=Cambridge | publisher=Cambridge University Press | year=1973 | isbn=0-521-20016-4| title-link=The Large Scale Structure of Space-Time }}</ref> In the context of [[general relativity]], a [[spacetime]] manifold is space orientable if, whenever two right-handed observers head off in rocket ships starting at the same spacetime point, and then meet again at another point, they remain right-handed with respect to one another. If a spacetime is time-orientable then the two observers will always agree on the direction of time at both points of their meeting. In fact, a spacetime is time-orientable if and only if any two observers can agree which of the two meetings preceded the other.<ref>{{cite journal |first=Mark J. |last=Hadley (|year=2002) [|url=http://www.iop.org/EJ/article/0264-9381/19/17/308/q21708.pdf?request-id=49d1e985-bf89-4203-b020-48367545e3c0 |title=The Orientability of Spacetime], |journal=[[Classical and Quantum Gravity]] |volume=19: 4565-4571|issue=17 [https://arxiv|pages=4565–71 |doi=10.org1088/abs/gr0264-qc9381/0202031v419/17/308 arXiv:|citeseerx=10.1.1.340.8125 |arxiv=gr-qc/0202031v4]}}</ref>