Manifold with an orientation-reversing self-homeomorphism

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Definition

A connected orientable manifold is termed a manifold with an orientation-reversing self-homeomorphism if there is a homeomorphism from the manifold to itself that, if applied to any orientation of the manifold, produces the opposite orientation.

For a compact connected orientable manifold

For a compact connected orientable manifold, a self-homeomorphism is orientation-reversing if it induces the multiplication by $-1$ map on the top homology, or equivalently, sends a fundamental class to its negative. A manifold with an orientation-reversing self-homeomorphism is thus a manifold for which there exists a self-homeomorphism with such an effect.

Examples

Below are some examples of compact connected orientable manifolds and whether or not they have orientation-reversing self-homeomorphisms:

Manifold or family of manifolds	Dimension in terms of parameter	For what values of the parameter does it have an orientation-reversing self-homeomorphism?	Proof/explanation
sphere $S^{n}$, $n\geq 1$	$n$	all	reflection in any one coordinate, keeping the other coordinates fixed.
real projective space $\mathbb {P} ^{n}(\mathbb {R} )$, $n$ odd	$n$	all odd $n$ (note: for $n$ even, the manifold isn't orientable to begin with, so the question doesn't make sense)	use the reflection in any one coordinate orientation-reversing self-homeomorphism of the sphere, descend it to real projective space (this is possible because the operation commutes with the antipodal map).
complex projective space $\mathbb {P} ^{n}(\mathbb {C} )$	$2n$	all odd $n$	complex projective space has orientation-reversing self-homeomorphism iff it has odd complex dimension
compact orientable surface, genus $g\geq 0$	2	all $g$	compact orientable surface has orientation-reversing self-homeomorphism

Facts

• Given two connected orientable manifolds $M_{1}$ and $M_{2}$, if either of them possesses an orientation-reversing self-homeomorphism, so does the product $M_{1}\times M_{2}$. The trick is to take the Cartesian product of the orientation-reversing self-homeomorphism on one manifold and the identity map on the other.