# Applying path-connectedness

*This is a survey article about applying the concept/definition/theorem*: path-connectedness

This article is about how one can use a point set-topological fact that a topological space (possibly with a lot of additional structure) is path-connected.

The hypothesis of being path-connected is in general stronger than the hypothesis of being connected, although for locally path-connected spaces, such as manifolds, the two hypotheses are equivalent. In certain situations, connectedness is a more useful hypothesis, while in other situations, it is more useful to use path-connectedness. Refer the article on applying connectedness.

## Contents

## Constructing paths and homotopies

The most useful and direct application of path-connectedness is to construct paths, which can be used to define homotopies and change basepoints.

### Basepoint-independence of homotopy groups

The homotopy groups of a topological space are usually defined with respect to a basepoint; in order to prove that the homotopy groups at two basepoints are isomorphic, we use a path between them to give an isomorphism. In particular, the fundamental groups at any two basepoints in a path-connected space are the same. Moreover, if is a path-connected space, and is a space with nondegenerate basepoint , then the homotopy classes of based maps from to can be identified for different choices of basepoint .

`Further information: Actions of the fundamental group`

### Multiplication maps in a group

In a path-connected topological group, the multiplication map by any group element is homotopy-equivalent to the identity map. The *homotopy* in this case is given by the path joining the group element to the identity element. In particular, any map given by an algebraic formula is homotopy-equivalent to a power map.

This fact is useful, for instance, in showing that any compact connected nontrivial Lie group has zero Euler characteristic. The idea is that the fixed-point free map of left multiplication by a group element, is homotopy-equivalent to the identity map.

### Constructing more complicated homotopies

A homotopy can be viewed as a kind of generalized path, and we can often use a path, or a loop, as the starting point of a homotopy. The typical additional concepts we need are those of a cofibration, and we often need to use tricks such as the two sides lemma or three sides lemma.