Manifold with an orientation-reversing self-homeomorphism: Difference between revisions

Latest revision as of 18:29, 27 July 2011

This article defines a property of topological spaces: a property that can be evaluated to true/false for any topological space|View a complete list of properties of topological spaces

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 $P^{n} (R)$ , $n$ 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 $P^{n} (C)$	$2 n$	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.

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 | complex projective space <math>\mathbb{P}^n(\mathbb{C})</math> || <math>2n</math> || all odd <math>n</math> || [[complex projective space has orientation-reversing self-homeomorphism iff it has odd complex dimension]]
 |-
-| [[compact orientable surface]], genus <math>g \ge 0</math> || 2 || all <math>g</math> || [[compact orientable surface possesses orientation-reversing self-homeomorphism]]
+| [[compact orientable surface]], genus <math>g \ge 0</math> || 2 || all <math>g</math> || [[compact orientable surface has orientation-reversing self-homeomorphism]]
 |}