Loop space of a based topological space

As a (based) 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 $$X$$).
 * The basepoint of this topological space is chosen as the constant loop that stays at the point $$x_0$$.

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:

$$t \mapsto \lbrace \begin{array}{rl} f_1(2t), & 0 \le t < 1/2 \\ f_2(2t - 1), & 1/2 \le t \le 1 \\\end{array}$$

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:

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 $$\Omega(X,x_0)$$.

Relationship with fundamental group
The fundamental group of a based topological space $$(X,x_0)$$, denoted $$\pi_1(X,x_0)$$, can be identified with the set of path components of $$Omega(X,x_0)$$. In other words:

$$\! \pi_1(X,x_0) \cong \pi_0(\Omega(X,x_0))$$

The identification is as follows: we know that the elements of $$\pi_1(X,x_0)$$ are precisely the homotopy classes of loops in $$X$$ based at $$x_0$$. The homotopy classes, in turn, are precisely the path components of $$\Omega(X,x_0)$$, 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 $$\Omega(X,x_0)$$ induces a monoid structure on the space $$\pi_0(\Omega(X,x_0))$$ 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:

$$\! \pi_k(X,x_0) \cong \pi_{k-1}(\Omega(X,x_0))$$

In fact, for $$k \ge 2$$, 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 $$\Omega^k(X,x_0)$$. This is obtained by iterating the loop space construction $$k$$ times. Note that at each stage, the new basepoint is chosen as the constant loop taking the old basepoint as its value.