nLab relative loop space




topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory


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topological homotopy theory

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A pointed topological pair is a pointed topological space X=(X,x 0)Top *X=(X,x_0)\in Top_* equipped with a topological subspace AXA\subset X containing the base point x 0x_0.

Let I=[0,1]I=[0,1] be the unit closed interval and for NN\in\mathbb{N} let J NI N{0}×I N1J^N \coloneqq \partial I^N-\{0\}\times I^{N-1}.


The relative loop space of a pointed topological pair (X,A)(X,A) is the space of continuous maps of the form (I N,I N,J N)(X,A,x 0)(I^N,\partial I^N,J^N)\rightarrow (X,A,x_0), denoted Ω N(X,A)\Omega^N(X,A).

For A={x 0}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.


For N1N\geq 1 the NN-th relative homotopy set of a topological pair (X,A)(X,A), denoted π N(X,A)\pi_N(X,A) is defined as the set of connected components of the associated relative loop space, ie π N(X,A)π 0Ω N(X,A)\pi_N(X,A)\coloneqq \pi_0\Omega^N(X,A).

If N2N\geq 2 then π N(X,A)\pi_N(X,A) is a group, which is abelian if N3N\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 * Top^\to_* of continuous pointed maps equipped with the projective model structure on functors. If we start with the Quillen model structure on TopTop, the cofibrant objects in Top * 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.


For N1N\geq 1 the relative NN-loop space functor is the right derivable functor

Ω rel N:Top * Top *,Ω rel N(ι:AX)(A× XX I) 𝕊 N1. \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.

Last revised on March 29, 2023 at 01:41:49. See the history of this page for a list of all contributions to it.