The definitions I know of orientability of manifolds are in terms of tangent spaces. However, for example in this answer there is mention of orientation preserving homeomophisms (between orientable complex manifolds).
My first thought would be something like instead of considering a basis of tangent vectors (or their determinant), to integrate these to consider a set of short curves emanating from a point on the manifold (e.g. by mapping curves in a coordinate chart), and considering the ordering of their image under the homeomorphism. It is not clear to me that this would work however, since a curve through a point could get folded in such a way that one end of it would give you one ordering, but the other a different one, or so it seems. Probably there could be other kinds of pathological behaviour as well.
In any case my questions are
- How can we define if a homeomorphism between oriented smooth manifolds is orientation-preserving?
- More generally, can we speak of orientability of topological manifolds without relying on any differentiable structure? Or even for more general topological spaces?
Thanks!
Best Answer
Orientation can be defined on manifolds using singular homology theory. The easiest case is where $M$ is a compact $n$-manifold. Then $H_n(M;\Bbb Z)$ is isomorphic to $\Bbb Z$ when $M$ is orientable, and is zero when $M$ is non-orientable. An orientation of $M$ is a choice of a generator of $H_n(M;\Bbb Z)$. A homeomorphism $f:M\to M'$ is orientation-preserving if $f^*$ maps your favoured generator of $H_n(M;\Bbb Z)$ to your favoured generator of $H_n(M';\Bbb Z)$.
More generally one can use local homology groups; these are relative homology groups $H_n(M,M-\{x\};\Bbb Z)$ where $x\in M$. These determine whether or not a map between manifolds is orientation-preserving at a given point.
For details, see texts in algebraic topology, for instance Hatcher's.