# Is the Empty set an orientable manifold?

The empty set can be regarded as an object in the category of smooth manifolds, at least for technical considerations.

Is the empty set an orientable manifold?

-
Just a note: The authors of this book books.google.no/… stipulate in the footnote on page 327 that the empty set is orientable. –  Fredrik Meyer Oct 27 '12 at 14:26

EDIT: This is wrong. See Henning Makholm's answer below.

An $n$-manifold is orientable if and only if it has a nonzero differential form taking $n$ arguments. The empty manifold has no nonzero forms whatsoever, so it is not orientable in any dimension.

-
I'm not sure if I like this answer. What about using the empty map? The form only has to be non-zero on every point of the manifold which is trivally satisfied for the emptyset. –  M.B. Oct 27 '12 at 13:39
The relevant condition is not at the differential form must be different from the everywhere zero form, but that at each point in the manifold, the differential form's restriction to that point's tangent space must be nonzero. This is true of the empty map, because there are no points to check it for. –  Henning Makholm Oct 27 '12 at 13:41
But what if we view the emptyset as a closed manifold. It most certainly does not have $\mathbb{Z}$ as its top homology group. This gives that the emptyset is not orientable for any $n$. –  M.B. Oct 27 '12 at 13:43
@commenter: Hm, oops. I'll have to claim that the empty set is disconnected then. But that it certainly is, because the set of its connected components has cardinality $\ne 1$. :-) –  Henning Makholm Oct 27 '12 at 13:57