Loop space of a based topological space

Definition

As a topological space

Suppose $(X,x_{0})$ is a based topological space, i.e., $X$ is a topological space and $x_{0}$ is a point in $X$. The loop space of $(X,x_{0})$, denoted $\Omega (X,x_{0})$, is defined as follows:

• As a set, it is the set of all continuous maps from the based unit circle $S^{1}$ (i.e., the unit circle with a fixed basepoint) to $(X,x_{0})$. In other words, all these maps send the chosen basepoint of $S^{1}$ to $x_{0}$.
• 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 $S^{1}$ to $X$).

As a H-space=

For convenience, we treat the unit circle as the quotient of the closed unit interval $[0,1]$ under the identification of $0$ and $1$, and the identified $0\sim 1$ point is treated as the basepoint.

The loop space $\Omega (X,x_{0})$ admits a multiplicative structure by concatenation and reparametrization, where, for loops $f_{1}$ and $f_{2}$, we define $f_{1}*f_{2}$ as the loop:

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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 $\Omega (X,x_{0})$.