Definition of orientation-preserving homeomorphism

Question

What is the definition of an orientation-preserving homeomorphism for a topological manifold $M$? If no such notion exists, then what is the definition of an orientation-preserving homeomorphism of $\mathbb{R}^n$?

I have seen these terms pop-up, but I cannot seem to find an actual definition of them.

Answer 1

On any topological $n$-manifold $M$, define an orientation of $M$ to be a function $\mu$ defined on $M$ such that for each input $x \in M$, the output $\mu(x)$ is one of the two generators of the infinite cyclic group $H_n(M,M-x)$, and the following property holds: for each embedded open $n$-ball $B \subset M$ there exists a generator $\mu_B$ of the infinite cyclic group $H_n(M,M-B)$ such that for each $x \in B$ the inclusion induced homomorphism $H_n(M,M-B) \mapsto H_n(M,M-x)$ maps $\mu_B$ to $\mu(x)$.

Now one proves

• Theorem (and definition): For every topological manifold $M$ exactly one of two possibilities holds: either $M$ has exactly two orientations, in which case we say that $M$ is orientable; or $M$ has no orientations, in which case we say that $M$ is nonorientable.

This theorem is one of the preliminary steps to the proof of Poincare Duality; see for example Hatcher's book "Algebraic Topology". In fact, proving that the relative homology groups $H_n(M,M-B)$ and $H_n(M,M-x)$ are infinite cyclic is also one of the preliminary steps.

Now to your question.

Let $M$ be a connected topological manifold.

If $M$ is nonorientable, then it makes no sense to ask whether a homeomorphism of $f$ preserves orientation, and the whole concept of "preserving orientation" is undefined for $M$.

If on the other hand $M$ is orientable then to say that a homeomorphism $f : M \to M$ is orientation preserving means that for either of the two orientations $\mu$ of $M$, and for any $x \in M$, the induced isomorphism $f : H_n(M,M-x) \to H_n(M,M-f(x))$ takes $\mu(x)$ to $\mu(f(x))$.