free loop 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.

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.


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.

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.


Revised on February 23, 2017 13:07:53 by Urs Schreiber (