If is a compact orientable-dimensional manifold, then embeds in ( for not a power of 2 the orientability condition is superfluous ).
22.
Similarly, the notion of orientability of a polyhedron is insufficient to characterise the surface twistings of toroidal polytopes, and this led to the use of torsion coefficients.
23.
Similarly, the notion of orientability of a polyhedron is insufficient to characterise the surface twistings of toroidal 4-polytopes, and this led to the use of torsion coefficients.
24.
(note the final B _ n is missing the superscript " 1; this flip, as compared to the orientable case, being the source of the non-orientability ).
25.
Weeks doesn't discuss the implications, if any, of non-orientability, i . e . the existence of closed paths in space such that a traveler comes back with reversed chirality.
26.
More concretely, triviality of the determinant bundle \ Omega ^ n ( M ) is equivalent to orientability, and a line bundle is trivial if and only if it has a nowhere-vanishing section.
27.
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.
28.
These homology classes agree on chart overlaps because of the orientability of the atlas, so they may be patched together to get a class in H _ n ( M; \ mathbf { Z } ).
29.
Two smooth surfaces are diffeomorphic if and only if they are homeomorphic . ( The analogous result does not hold for higher-dimensional manifolds . ) Thus closed surfaces are classified up to diffeomorphism by their Euler characteristic and orientability.
30.
Two smooth surfaces are diffeomorphic if and only if they are homeomorphic . ( The analogous result does not hold for manifolds of dimension greater than three . ) It follows that closed surfaces are classified up to diffeomorphism by their Euler characteristic and orientability.
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