In Lorentzian geometry, there are two kinds of orientability : space orientability and time orientability.
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Thus, for surfaces, the M�bius strip may be considered the source of all non-orientability.
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Stated another way, the only characteristic class of a 3-manifold is the obstruction to orientability.
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Some key criteria include the " simply connected " property and orientability ( see below ).
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:Real projective space now addresses orientability after talk ) 06 : 17, 27 May 2007 ( UTC)
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Orientability of a manifold implies that its top homology group is isomorphic to "'Z " '.
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A Thom isomorphism theorem for a homology theory is now accepted as the generalized notion of orientability for a homology theory.
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In this setting, various equivalent formulations of orientability can be given, depending on the desired application and level of generality.
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The most familiar example is orientability : some manifolds are orientable, some are not, and orientable manifolds admit 2 orientations.
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At the heart of all the above definitions of orientability of a differentiable manifold is the notion of an orientation preserving transition function.
