# nLab free loop space

Contents

### Context

#### Mapping space

internal hom/mapping space

# Contents

## Idea

The free loop space $\mathcal{L}X$ 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 $x_0\in X$.

(Regarded as a homotopy type the concept generalizes to other contexts of homotopy theory, see at free loop space object for more.)

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

## Definition

### Explicit description

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

If we work in a category of based spaces, then still the topological space $Map(S^1,X)$ is in the non-based sense but 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 $L X$ of $X$ is defined as the free loop space object of $X$ formed in the (∞,1)-category Top.

### Cohomology of $\mathcal{L}X$ and Hochschild homology of $X$

Let $X$ be a simply connected topological space.

The ordinary cohomology $H^\bullet$ of its free loop space is the Hochschild homology $HH_\bullet$ of its singular chains $C^\bullet(X)$:

$H^\bullet(\mathcal{L}X) \simeq HH_\bullet( C^\bullet(X) ) \,.$

Moreover the $S^1$-equivariant cohomology of the loop space, hence the ordinary cohomology of the cyclic loop space $\mathcal{L}X/^h S^1$ is the cyclic homology $HC_\bullet$ of the singular chains:

$H^\bullet(\mathcal{L}X/^h S^1) \simeq HC_\bullet( C^\bullet(X) )$

(Loday 11)

If the coefficients are rational, and $X$ is of finite type then this may be computed by the Sullivan model for free loop spaces, see there the section on Relation to Hochschild homology.

In the special case that the topological space $X$ carries the structure of a smooth manifold, then the singular cochains on $X$ are equivalent to the dgc-algebra of differential forms (the de Rham algebra) and hence in this case the statement becomes that

$H^\bullet(\mathcal{L}X) \simeq HH_\bullet( \Omega^\bullet(X) ) \,.$
$H^\bullet(\mathcal{L}X/^h S^1) \simeq HC_\bullet( \Omega^\bullet(X) ) \,.$

This is known as Jones' theorem (Jones 87)

An infinity-category theoretic proof of this fact is indicated at Hochschild cohomology – Jones’ theorem.

## References

• David Chataur, Alexandru Oancea, Basics on free loop spaces, Chapter I in: Janko Latchev, Alexandru Oancea (eds.): Free Loop Spaces in Geometry and Topology, IRMA Lectures in Mathematics and Theoretical Physics 24, EMS 2015 [ISBN:978-3-03719-153-8]

• Kathryn Hess, Free loop spaces in topology and physics, talk at Meeting of Edinburgh Math. Soc. Glasgow (14 Nov 2008) [pdf, pdf]

In the context of Hochschild homology (and cyclic homology):

reviewed in:

Last revised on January 21, 2024 at 21:17:42. See the history of this page for a list of all contributions to it.