nLab
free loop space

Contents

Idea

The free loop space X\mathcal{L}X of a topological space XX (based or not) is the space of all loops in XX. This is in contrast to the based loop space of a based space XX for which the loops are at the fixed base point x 0Xx_0\in X.

The free loop space carries a canonical action (infinity-action) of the circle group. The homotopy quotient by that action (X)/S 1\mathcal{L}(X)/S^1 (the “cyclic loop space”) contains what is known as the twisted loop space of XX.

Definition

Explicit description

For XX a topological space, the free loop space LXL X is the topological space Map(S 1,X)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 Map(S 1,X)Map(S^1,X) is in the non-based sense but has a distinguished point which is the constant map tx 0t\mapsto x_0 where x 0x_0 is the base point of XX.

General abstract description

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

In rational homotopy theory

See at Sullivan model of free loop space.

References

Revised on February 14, 2017 05:00:44 by Urs Schreiber (83.208.22.80)