nLab
cyclic loop space

Contents

Idea

Any free loop space β„’X\mathcal{L}X has a canonical action (infinity-action) of the circle group S 1S^1. The homotopy quotient β„’(X)/S 1\mathcal{L}(X)/S^1 of this action might be called the cyclic loop space of XX.

If X=Spec(A)X = Spec(A) is an affine variety regarded in derived algebraic geometry, then π’ͺ(β„’Spec(A))\mathcal{O}(\mathcal{L}Spec(A)) is the Hochschild homology of AA and π’ͺ((β„’Spec(A))/S 1)\mathcal{O}((\mathcal{L}Spec(A))/S^1) the corresponding cyclic homology, see the discussion at Hochschild cohomology.

If X=Y//GX = Y//G is the homotopy quotient of a topological space by a topological group action, regarded as a locally constant ∞\infty-stack, so that the S 1S^1-action on β„’(X//G)\mathcal{L}(X//G) is an Bβ„€B \mathbb{Z}-action, then the restriction of the cyclic loop space to the constant loops β„’ constY//Gβ†’β„’(Y//G)\mathcal{L}_{const}Y//G \to \mathcal{L}(Y//G) has been called the twisted loop space in (Witten 88). This terminology has been widely adopted, for example in the context of the transchromatic character map (Stapleton 11)

Properties

As right base change along *β†’BS 1\ast \to \mathbf{B} S^1

The cyclic loop space β„’X/S 1\mathcal{L}X/S^1 is equivalently the right base change/dependent product along the canonical point inclusion *β†’BS 1\ast \to B S^1 (this prop.). See also at double dimensional reduction.

Ordinary cohomology of β„’X/S 1\mathcal{L}X/S^1 on cyclic cohomology of XX

Let XX be a simply connected topological space.

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

H β€’(β„’X)≃HH β€’(C β€’(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 β„’X/ hS 1\mathcal{L}X/^h S^1 is the cyclic homology HC β€’HC_\bullet of the singular chains:

H β€’(β„’X/ hS 1)≃HC β€’(C β€’(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 β€’(β„’X)≃HH β€’(Ξ© β€’(X)). H^\bullet(\mathcal{L}X) \simeq HH_\bullet( \Omega^\bullet(X) ) \,.
H β€’(β„’X/ hS 1)≃HC β€’(Ξ© β€’(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.

Rational Sullivan model

See at Sullivan model for free loop space

References

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

  • Edward Witten, The index of the Dirac operator in loop space. In Elliptic curves and modular forms in algebraic topology (Princeton, NJ, 1986), volume 1326 of Lecture Notes in Math., pages 161–181. Springer, Berlin, 1988.

  • Jean-Louis Loday, Free loop space and homology (arXiv:1110.0405)

  • Nathaniel Stapleton, Transchromatic generalized character maps, Algebr. Geom. Topol. 13 (2013) 171-203 (arXiv:1110.3346)

Last revised on March 14, 2017 at 12:58:14. See the history of this page for a list of all contributions to it.