# Loop space of a based topological space

## Contents

## Definition

### As a (based) topological space

Suppose is a based topological space, i.e., is a topological space and is a point in . The **loop space** of , denoted , is defined as follows:

- As a set, it is the set of all continuous maps from the based unit circle (i.e., the unit circle with a fixed basepoint) to . In other words, all these maps send the chosen basepoint of to .
- The topology on this set is the compact-open topology (or rather, the subspace topology from the compact-open topology on all continuous maps from to ).
- The basepoint of this topological space is chosen as the
*constant*loop that stays at the point .

## As a H-space=

For convenience, we treat the unit circle as the quotient of the closed unit interval under the identification of and , and the identified point is treated as the basepoint.

The loop space admits a multiplicative structure by *concatenation and reparametrization*, where, for loops and , we define as the loop:

This multiplicative structure is continuous, making the loop space a topological magma. However, it is *not* a topological monoid, because the multiplication is not strictly associative and does not have a strict identity element. Instead, it is a H-space, in the sense that the multiplication is associative up to homotopy and there is an element that works as an identity element up to homotopy:

Condition | Proof |
---|---|

Associativity up to homotopy | homotopy between composites associated in different ways |

Identity element up to homotopy | homotopy between loop and composite with constant loop |

What is important is not just that there exist individual homotopies for the associativity of each triple, but that these homotopies vary continuously, so that we get a homotopy at the level of the topological space .

## Relation with other constructs

### Relationship with fundamental group

The fundamental group of a based topological space , denoted , can be identified with the set of path components of . In other words:

The identification is as follows: we know that the elements of are precisely the homotopy classes of loops in based at . The *homotopy classes*, in turn, are precisely the path components of , because a homotopy of based loops is a path in the space of based loops under the compact-open topology.

In addition to the identification as a *set*, we can also make the identification as a group. The left side has a group structure under concatenation. On the right side, the H-space structure of induces a monoid structure on the space of its path components. That monoid turns out to be a group, and the identification is a group isomorphism.

### Relationship with higher homotopy groups

More generally, we have the following relationship:

In fact, for , both sides are naturally abelian groups, and the natural identification is an isomorphism of abelian groups.

### Iterated loop spaces

We can also consider the iterated loop space . This is obtained by iterating the loop space construction times. Note that at each stage, the new basepoint is chosen as the *constant* loop taking the old basepoint as its value.