# Weakly contractible space

From Topospaces

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

This is a variation of contractible space. View other variations of contractible space

## Contents

## Definition

### Equivalent definitions in tabular format

A nonempty topological space is said to be **weakly contractible** if it satisfies the following equivalent conditions. The empty space is generally excluded from consideration when considering the question of weak contractibility.

As we see below, each of the definitions (implicitly or explicitly) implies that the space is a path-connected space.

No. | Shorthand | A topological space is termed weakly contractible if ... | A topological space is termed weakly contractible if ... |
---|---|---|---|

1 | weakly homotopy-equivalent to a point | it is weakly homotopy-equivalent to a one-point space. Note that the definition in particular implies that the space is a path-connected space. | is path-connected, and it is in the same equivalence class as the one-point space under weak homotopy equivalence. |

2 | weakly homotopy-equivalent to a point (unique map to the point) | it is path-connected and the unique map from it to the one-point space is a weak homotopy equivalence of topological spaces (regardless of choice of basepoint). | (fix any point ; the truth of this fact does not depend on the choice of ) is path-connected, and the unique map to the one-point space (which is continuous by definition) induces isomorphisms on all the homotopy groups. |

3 | weakly homotopy-equivalent to a point (arbitrary choice of basepoint) | it is path-connected and any map from a one-point space to it induces a weak homotopy equivalence of topological spaces | (fix any point ; the truth of this fact does not depend on the choice of ) Suppose is the (continuous by definition) map from the one-point space to that sends the point to . Then, induces isomorphisms on all the homotopy groups. |

4 | all homotopy groups are trivial | it is path-connected (i.e., its set of path components has size one) and all its homotopy groups are trivial. | (fix any point ; the truth of this fact does not depend on the choice of ) is path-connected and is the trivial group for all positive integers . |

5 | any map from a sphere is nullhomotopic | it is path-connected and any continuous map from a sphere (of any finite dimension) to it is nullhomotopic. | is path-connected, and any continuous map for any is nullhomotopic, i.e., we can construct a homotopy from that map to a map that sends to a single point. Note that restricting ourselves to homotopies that fix a basepoint does not change the strength of the definition. |

6 | any map from a connected CW-space is nullhomotopic | it is path-connected and any continuous map from a connected CW-space (the underlying space of a CW-complex) to it is nullhomotopic. | is path-connected and for any connected CW-space and any continuous map , is nullhomotopic. Note that restricting ourselves to homotopies that fix a basepoint does not change the strength of the definition. |

7 | any map from a connected polyhedron is nullhomotopic | it is path-connected and any continuous map from a connected polyhedron (the geometric realization of a simplicial complex) to it is nullhomotopic. | is path-connected and for any connected polyhedron and any continuous map , is nullhomotopic. Note that restricting ourselves to homotopies that fix a basepoint does not change the strength of the definition. |

8 | any map from a connected manifold is nullhomotopic | it is path-connected and any continuous map from a connected manifold to it is nullhomotopic. | is path-connected and for any connected manifold and any continuous map , is nullhomotopic. Note that restricting ourselves to homotopies that fix a basepoint does not change the strength of the definition. |

9 | any map from a manifold is nullhomotopic | it is path-connected and any continuous map from a manifold to it is nullhomotopic. | is path-connected and for any manifold and any continuous map , is nullhomotopic. Note that restricting ourselves to homotopies that fix a basepoint does not change the strength of the definition. |

## Relation with other properties

### Stronger properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

contractible space | admits a contracting homotopy | (direct from definition) | weakly contractible not implies contractible (however, Whitehead's theorem says that weakly contractible equals contractible when we restrict to CW-spaces) | |FULL LIST, MORE INFO |

topologically convex space | homeomorphic to a convex subset of Euclidean space | (via contractible) | (via contractible) | Contractible space|FULL LIST, MORE INFO |

### Weaker properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

acyclic space | all the homology groups are the same as those of a point | weakly contractible implies acyclic | |FULL LIST, MORE INFO | |

rationally acyclic space | all the homology groups over the rationals are the same as those of a point | Acyclic space|FULL LIST, MORE INFO | ||

simply connected space | the fundamental group is trivial | |FULL LIST, MORE INFO | ||

simple space | the fundamental group is abelian and it acts trivially on all the higher homotopy groups | Simply connected space|FULL LIST, MORE INFO | ||

path-connected space | any two points can be connected by a path | |FULL LIST, MORE INFO |

## Metaproperties

### Products

This property of topological spaces is closed under taking arbitrary products

View all properties of topological spaces closed under products

Since the homotopy group of the product of two spaces is the product of their homotopy groups, the product of two weakly contractible spaces is again weakly contractible.

### 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

Any retract, and more generally, any homotopically injective subspace of a weakly contractible space is again weakly contractible.