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 $S^{n},n\geq 1$	$n$	No	The antipodal map is a fixed-point-free self-map (in fact, it's a self-homeomorphism without fixed points).
real projective space $\mathbb {P} ^{n}(\mathbb {R} ),n\geq 1$	$n$	Even $n$: Yes Odd $n$: No	See real projective space has fixed-point property iff it has odd dimension. For $n$ even, follows from rationally acyclic compact polyhedron has fixed-point property. For $n$ odd, we use the mapping $[a_{1}:a_{2}:a_{3}:a_{4}:\dots :a_{n}:a_{n+1}]\mapsto [-a_{2}:a_{1}:-a_{4}:a_{3}:\dots :-a_{n+1}:a_{n}]$. Note that the map has no fixed points because solving the fixed point condition gives $a_{1}^{2}+a_{2}^{2}=a_{3}^{2}+a_{4}^{2}=\dots =a_{n}^{2}+a_{n+1}^{2}=0$, forcing $a_{1}=a_{2}=\dots =a_{n}=a_{n+1}=0$.
complex projective space $\mathbb {P} ^{n}(\mathbb {C} ),n\geq 1$	$2n$	Even $n$: Yes Odd $n$: No	See complex projective space has fixed-point property iff it has odd complex dimension. Explicitly, for $n$ odd, we use the mapping $[a_{1}:a_{2}:a_{3}:a_{4}:\dots :a_{n}:a_{n+1}]\mapsto [-{\overline {a_{2}}}:{\overline {a_{1}}}:-{\overline {a_{4}}}:{\overline {a_{3}}}:\dots :-{\overline {a_{n+1}}}:{\overline {a_{n}}}]$. Note that the map has no fixed points because solving the fixed point condition gives $|a_{1}|^{2}+|a_{2}|^{2}=|a_{3}|^{2}+|a_{4}|^{2}=\dots =|a_{n}|^{2}+|a_{n+1}|^{2}=0$, forcing $a_{1}=a_{2}=\dots =a_{n}=a_{n+1}=0$.
compact orientable surface of genus $g\geq 0$	2	$g=0,1$: No $g\geq 2$: Yes (need to verify)
torus $T^{n},n\geq 1$, product of $n$ circles	$n$	No	Follows from the fact that it's a nontrivial Lie group, so multiplication by a non-identity element. Also, we can use the coordinate-wise antipodal map.
product of spheres $S^{m_{1}}\times S^{m_{2}}\times \dots S^{m_{r}}$	$m_{1}+m_{2}+\dots +m_{r}$	No	Take the antipodal map for each coordinate sphere.

Facts

Fixed-point theorems

• 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.

Products

The product of any topological space that does not satisfy the fixed-point property with any nonempty topological space gives a space that does not satisfy the fixed-point property.

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