nLab
free loop space

Idea

The free loop space of a topological space $X$ (based or not) is the space of all loops in $X$. This is in contrast to the based loop space of a based space $X$ for which the loops are at the fixed base point $x_0\in X$.

Definition

Explicit description

For $X$ a topological space, the free loop space $LX$ is the topological space $\mathrm{Map}(S^1,X)$ of continuous maps in compact-open topology.

If we work in a category of based spaces, then still the topological space $\mathrm{Map}(S^1,X)$ is in the non-based sense bit it itself has a distinguished point which is the constant map $t\mapsto x_0$ where $x_0$ is the base point of $X$.

General abstract description

If $X$ is a topological space, the free loop space $LX$ of $X$ is defined as the free loop space object of $X$ formed in the (∞,1)-category Top.

Related entries and literature

• loop space object, free loop space object,

  • delooping
  • loop space, free loop space, derived loop space

• suspension object * suspension

• Kathryn Hess, Free loop spaces in topology and physics, pdf, slides from Meeting of Edinburgh Math. Soc. Glasgow, 14 Nov 2008

• D. Ben-Zvi, D. Nadler, Loop spaces and connections, arxiv/1002.3636