nLab free loop space

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.

In rational homotopy theory

See at Sullivan model of free loop space.

Examples

Example

For $G$ a topological group, the free loop space of its underlying topological space is homeomorphic to the product space of $G$ with its based loop space:

$$\mathcal{L}G \simeq G \times \Omega G \mathrlap{\,,}$$

where the base point of $G$ is taken to be the neutral element.

Moreover, this homeomorphism is also a group homomorphism with respect to the pointwise group operation on $\mathcal{L}G$ and $\Omega G$ (cf. at loop group) and the following semidirect product group structure on $G \times \Omega G$:

$$(g_1,\ell_1) (g_2, \ell_2) \coloneqq \Big( g_1 g_2, \big( t \mapsto g_2^{-1} \ell_1(t) g_2 \ell_2(t) \big) \Big) \mathrlap{\,.}$$

Proof

Let the loops be parameterized over $[0,1]$. Consider the maps

$$\begin{array}{ccc} \mathcal{L} G &\xrightarrow{\phantom{--} \Phi \phantom{--}}& G \times \Omega G \\ \gamma &\mapsto& \Big( \gamma(0), \big( t \mapsto \gamma(0)^{-1}\gamma(t) \big) \Big) \end{array}$$

and

$$\begin{array}{ccc} G \times \Omega G &\xrightarrow{\phantom{--} \overline{\Phi} \phantom{--}}& \mathcal{L} G \\ (g, \ell) &\mapsto& \big( t \mapsto g \ell(t) \big) \end{array}$$

These maps are both continuous since the operations of forming products and inverses in $G$ are continuous, by assumption on $G$. So it remains to check that the maps are inverses of each other. Indeed, unwinding the definitions, we have:

$$\begin{aligned} \overline{\Phi}\big(\Phi(\gamma)\big) & = \overline{\Phi}\Big( \gamma(0), \big( t \mapsto \gamma(0)^{-1}\gamma(t) \big) \Big) \\ & = \big( t \mapsto \gamma(0) \gamma(0)^{-1}\gamma(t) \big) \\ & = \gamma \end{aligned}$$

and

$$\begin{aligned} \Phi\big(\overline{\Phi}(g,\ell)\big) & = \Phi\big( t \mapsto g \ell(t) \big) \\ & = \bigg( g \ell(0), \Big( t \mapsto \big(g \ell(0)\big)^{-1} g \ell(t) \Big) \bigg) \\ & = \Big( g , \big( t \mapsto \ell(t) \big) \Big) \\ & = (g, \ell) \mathrlap{\,.} \end{aligned}$$

Similarly, it is immediate to check that $\overline{\Phi}$ is a group homomorphism as claimed:

$$\begin{aligned} \overline{\Phi}\big( (g_1,\ell_1) (g_2, \ell_2) \big) & = \overline{\Phi}\bigg( \Big( g_1 g_2, \big( t \mapsto g_2^{-1} \ell_1(t) g_2 \ell_2(t) \big) \Big) \bigg) \\ & = \big( t \mapsto g_1 g_2 g_2^{-1} \ell_1(t) g_2 \ell_2(t) \big) \\ & = \big( t \mapsto g_1 \ell_1(t) g_2 \ell_2(t) \big) \\ & = \overline{\Phi}( g_1, \ell_1 ) \overline{\Phi}( g_2, \ell_2 ) \mathrlap{\,.} \end{aligned}$$

Example

For discussion of free loop spaces of classifying spaces see there.

Related entries

• loop space object, free loop space object,

  • delooping

  • loop space, free loop space, derived loop space

  • Sullivan model of free loop space

• Hochschild homology, cyclic homology, dihedral homology

• smooth loop space,

• free loop orbifold, inertia orbifold

• cyclic loop space, cyclic loop stack

• caloron correspondence

• suspension object

  • suspension

• fusion bundle

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):

• John D.S. Jones, Cyclic homology and equivariant homology, Invent. Math. 87, 403-423 (1987) (pdf, doi:10.1007/BF01389424)

reviewed in:

• Jean-Louis Loday, Cyclic Spaces and $S^1$-Equivariant Homology (doi:10.1007/978-3-662-21739-9_7)

  Chapter 7 in: Cyclic Homology, Grundlehren 301, Springer 1992 (doi:10.1007/978-3-662-21739-9)

• Jean-Louis Loday, Section 4 of: Free loop space and homology, Chapter 4 in: Janko Latchev, Alexandru Oancea (eds.): Free Loop Spaces in Geometry and Topology, IRMA Lectures in Mathematics and Theoretical Physics 24, EMS 2015 (arXiv:1110.0405, ISBN:978-3-03719-153-8)

See also:

• Manuel Rivera, Samson Saneblidze: A combinatorial model for the free loop space fibration, Bull. London Math. Soc. 50 (2018) 1085-1101 [doi:10.1112/blms.12202, arXiv:1712.02644]