# 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

## Contents

## 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 | No | The antipodal map is a fixed-point-free self-map (in fact, it's a self-homeomorphism without fixed points). | |

real projective space | Even : Yes Odd : No |
See real projective space has fixed-point property iff it has even dimension | |

complex projective space | Even : Yes Odd : No |
See complex projective space has fixed-point property iff it has even complex dimension. | |

quaternionic projective space | Even : Yes Odd : No |
See quaternionic projective space has fixed-point property iff it has even quaternionic dimension. | |

compact orientable surface of genus | 2 | No | |

torus , product of circles | 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 | No | Take the antipodal map for each coordinate sphere. |

### Manifolds with boundary

## Facts

### Fixed-point theorems

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 homology is where 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`