1. Smooth structures on an orientable manifold are usually counted modulo orientation-preserving smooth homeomorphisms. 2. On non-orientable manifolds , one may instead define the weaker notion of a density. 3. This is an orientable manifold with boundary, upon which " surgery " will be performed. 4. The singular homology and cohomology groups of a closed, orientable manifold are related by Poincar?duality. 5. A manifold is orientable if it has a consistent choice of connected orientable manifold has exactly two different possible orientations. 6. A true non-orientable manifold has " closed paths " that take travelers from R to / . 7. The most familiar example is orientability : some manifolds are orientable, some are not, and orientable manifolds admit 2 orientations. 8. See also orientable manifold ( the article may be a bit advanced, but it formally pinpoints some of the above observations ). 9. An orientable manifold has infinitely many volume forms, since multiplying a volume form by a non-vanishing function yields another volume form. 10. Both bundles are 2-manifolds, but the annulus is an orientable manifold while the M�bius band is a non-orientable manifold.
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