nLab relative loop space

Definition

A pointed topological pair is a pointed topological space $X=(X,x_0)\in Top_*$ equipped with a topological subspace $A\subset X$ containing the base point $x_0$.

Let $I=[0,1]$ be the unit closed interval and for $N\in\mathbb{N}$ let $J^N \coloneqq \partial I^N-\{0\}\times I^{N-1}$.

Definition

The relative loop space of a pointed topological pair $(X,A)$ is the space of continuous maps of the form $(I^N,\partial I^N,J^N)\rightarrow (X,A,x_0)$, denoted $\Omega^N(X,A)$.

For $A = \{x_0\}$ this reduces to the ordinary notion of (iterated) loop space.

The relative loop spaces allow us to define the relative homotopy groups of topological pairs.

Definition

For $N\geq 1$ the $N$-th relative homotopy set of a topological pair $(X,A)$, denoted $\pi_N(X,A)$ is defined as the set of connected components of the associated relative loop space, ie $\pi_N(X,A)\coloneqq \pi_0\Omega^N(X,A)$.

If $N\geq 2$ then $\pi_N(X,A)$ is a group, which is abelian if $N\geq 3$.

Homotopy theory of relative loop spaces

If we are interested in the homotopy theory of relative loop spaces (as in recognition of relative loop spaces) the above definition is not appropriate since there is no model category-structure on the category of topological pairs, as explained in this mathoverflow discussion.

The solution here is to work in the category $Top^\to_*$ of continuous pointed maps equipped with the projective model structure on functors. If we start with the Quillen model structure on $Top$, the cofibrant objects in $Top^\to_*$ are the inclusions of CW-pairs, and if we start with the mixed model structure we get the maps homotopy equivalent to those.

We can then define relative loop spaces as loop spaces of homotopy fibers.

Definition

For $N\geq 1$ the relative $N$-loop space functor is the right derivable functor

$$\Omega^N_{rel}:Top_*^\to \to Top_*, \qquad \Omega^N_{rel}(\iota:A\rightarrow X)\coloneqq (A\times_X X^I)^{\mathbb{S}^{N-1}}.$$

For inclusions of topological pairs the two definitions of relative loop spaces are naturally homeomorphic.

Related concepts

• loop space

• CW-pair