Stable homotopy theory
The free loop space of a topological space (based or not) is the space of all loops in . This is in contrast to the based loop space of a based space for which the loops are at the fixed base point .
The free loop space carries a canonical action (infinity-action) of the circle group. The homotopy quotient by that action (the “cyclic loop space”) contains what is known as the twisted loop space of .
For a topological space, the free loop space is the topological space of continuous maps in compact-open topology.
If we work in a category of based spaces, then still the topological space is in the non-based sense but has a distinguished point which is the constant map where is the base point of .
General abstract description
If is a topological space, the free loop space of is defined as the free loop space object of formed in the (∞,1)-category Top.
Cohomology of and Hochschild homology of
Let be a simply connected topological space.
The ordinary cohomology of its free loop space is the Hochschild homology of its singular chains :
Moreover the -equivariant cohomology of the loop space, hence the ordinary cohomology of the cyclic loop space is the cyclic homology of the singular chains:
If the coefficients are rational, and 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 carries the structure of a smooth manifold, then the singular cochains on are equivalent to the dgc-algebra of differential forms (the de Rham algebra) and hence in this case the statement becomes that
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.
Kathryn Hess, Free loop spaces in topology and physics, pdf, slides from Meeting of Edinburgh Math. Soc. Glasgow, 14 Nov 2008
John D.S. Jones, Cyclic homology and equivariant homology, Invent. Math. 87, 403-423 (1987) (pdf)
Jean-Louis Loday, Free loop space and homology (arXiv:1110.0405)
D. Ben-Zvi, D. Nadler, Loop spaces and connections, arxiv/1002.3636