nLab free loop space




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

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


Basic concepts

Universal constructions

Extra stuff, structure, properties


Basic statements


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topological homotopy theory

Mapping space



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.

(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 (X)/S 1\mathcal{L}(X)/S^1 (the “cyclic loop space”) contains what is known as the twisted loop space of XX.


Explicit description

For XX a topological space, the free loop space X\mathcal{L}X is the topological space Map(S 1,X)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)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.

Cohomology of X\mathcal{L}X and Hochschild homology of XX

Let XX be a simply connected topological space.

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

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

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

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

(Loday 11)

If the coefficients are rational, and XX 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 XX carries the structure of a smooth manifold, then the singular cochains on XX are equivalent to the dgc-algebra of differential forms (the de Rham algebra) and hence in this case the statement becomes that

H (X)HH (Ω (X)). H^\bullet(\mathcal{L}X) \simeq HH_\bullet( \Omega^\bullet(X) ) \,.
H (X/ hS 1)HC (Ω (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.



  • 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.