Fixed-point property

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 topological space is said to have the fixed-point property if every continuous map (not necessarily a self-homeomorphism) from the topological space to itself has a fixed point.

Relation with other properties

Stronger properties

• acyclic compact polyhedron (nonempty)
• rationally acyclic compact polyhedron (nonempty)

Examples

Manifolds without boundary

Manifold or family of manifolds	Dimension in terms of parameter	Does it satisfy the fixed-point property?	Proof/explanation
sphere $S^n,n\ge 1$	$n$	No	The antipodal map is a fixed-point-free self-map (in fact, it's a self-homeomorphism without fixed points).
real projective space ${\mathbb{P}}^n(\mathbb{R}),n\ge 1$	$n$	Yes if $n$ is even, No if $n=1,3$, what happens for other odd $n$?	For $n$ even, follows from rationally acyclic compact polyhedron has fixed-point property. For $n=1,3$, follows from the fact that it is a Lie group and thus multiplication by a non-identity element is a continuous self-map without fixed points.
complex projective space ${\mathbb{P}}^n(\mathbb{C}),n\ge 1$	$2n$	Yes if $n$ is even, what happens for odd $n$?	Fill this in later
compact orientable surface of genus $g\ge 0$	2	No if $g=0,1$, what happens for higher $g$?

Facts

• Lefschetz fixed-point theorem
• Brouwer fixed-point theorem

In general, we combine the Lefschetz fixed-point theorem with the structure of the cohomology ring of the space to determine whether or not it has the fixed-point property. For instance, we can show that complex projective space in even dimensions has the fixed-point property, by combining the Lefschetz fixed-point theorem with the fact that the trace on the $(2k{)}^{th}$ homology is $d^k$ where $d$ is the trace on the second homology.

Metaproperties

Retract-hereditariness

This property of topological spaces is hereditary on retracts, viz if a space has the property, so does any retract of it
View all retract-hereditary properties of topological spaces

Every retract of a space with the fixed-point property also has the fixed-point property. Further information: fixed-point property is retract-hereditary