nLab cyclic loop space

Idea

Any free loop space $\mathcal{L}X \,=\, Map(S^1,X)$ carries a canonical group action (infinity-action) of the circle group $S^1$ acting by rigid rotation on itself. The homotopy quotient $Cyc(X) \;\coloneqq\; \mathcal{L}(X) \sslash S^1$ of this action might be called the cyclic loop space of $X$:

If $X$ is a simply connected topological space, then the ordinary cohomology of $Cyc(X)$ is the cyclic cohomology of $X$ (Jones’ theorem [Jones (1987); Loday (1992), §7.2; Loday (2015)], see below).

Analogously, if $X = Spec(A)$ is an affine variety regarded in derived algebraic geometry, then $\mathcal{O}(\mathcal{L}Spec(A))$ is the Hochschild homology of $A$ and $\mathcal{O}((\mathcal{L}Spec(A))/S^1)$ the corresponding cyclic homology, see also the discussion at Hochschild cohomology.

But $Cyc(X)$ is also known as the string space of $X$ (Chataur 2005. 4.8.1, Bökstedt & Ottosen 05, p. 1).

If $X = Y\sslash G$ is the homotopy quotient of a topological space by a topological group action, regarded as a locally constant $\infty$-stack, then (a point set-model for) the free loop space $\mathcal{L}(Y \sslash G)$ has been called the twisted loop space in Witten 88; and this terminology is essentially carried over to the cyclification of its restriction to the constant loops Stapleton (2013), p. 2 in the context of the transchromatic character.

A candidate lift of this construction from plain homotopy types to smooth homotopy types, namely to orbifold stacks, is Huan's inertia orbifold-construction.

Properties

As right base change along $\ast \to \mathbf{B} S^1$

The cyclic loop space $\mathcal{L}X \sslash S^1$ is equivalently the right base change/dependent product along the canonical point inclusion $\ast \to B S^1$ (this prop.) into the delooping of $S^1$ (the classifying space of the circle group when realized in the homotopy theory of topological spaces). See also at double dimensional reduction (BMSS 19, Sec. 2.2, following FSS 18, Sec. 3).

Ordinary cohomology of $\mathcal{L}X \sslash S^1$ as cyclic cohomology 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 $\mathcal{L}X \sslash S^1$ is the cyclic homology $HC_\bullet$ of the singular chains:

$$H^\bullet(\mathcal{L}X \sslash S^1) \simeq HC_\bullet( C^\bullet(X) )$$

(Jones 87, Thm. A, review in Loday 92, Cor. 7.3.14, Loday 11, Sec. 4)

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

$$H^\bullet(\mathcal{L}X) \simeq HH_\bullet( \Omega^\bullet(X) ) \,.$$

$$H^\bullet(\mathcal{L}X \sslash S^1) \simeq HC_\bullet( \Omega^\bullet(X) ) \,.$$

This is known as Jones' theorem (Jones 87)

An infinity-category theoretic proof of this fact is indicated at Hochschild cohomology – Jones’ theorem.

Stable splitting

Proposition

For $X \in TopSpaces^{\ast/}$ a pointed topological space with reduced suspension denoted $\Sigma X$, the stabilization of its cyclic loop spaces splits as a direct sum of suspension spectra, hence the underlying infinite loop space of that as a product space, of the following form:

$$\Omega^\infty \Sigma^\infty \big( Maps(S^1, \Sigma X) \sslash S^1 \big) \;\; \simeq \;\; \big( \Omega^\infty \Sigma^\infty B S^1 \big) \times \underset{ k \in \mathbb{N}_+ }{\prod} \Omega^\infty \Sigma^\infty \big( (E (\mathbb{Z}/k))_+ \wedge_{\mathbb{Z}/k} X^{\wedge^n} \big)$$

In terms of the suspension spectra themselves:

$$\Sigma^\infty \Big( \underset{ \frac{ E S^1 \times_{S^1} Maps(S^1, \Sigma X) } { E S^1 \times_{S^1} \ast \,=\, B S^1 } }{ \underbrace{ Maps \big( S^1, \, \Sigma X \big) \wedge_{S^1} (E S^1)_+ } } \Big) \;\; \simeq \;\; \underset{ k \in \mathbb{N}_{\geq 1} }{\vee} \Sigma^\infty \big( (E \mathbb{Z}/k)_+ \wedge_{\mathbb{Z}/k} X^{\wedge^k} \big)$$

(Carlsson & Cohen 1987, Cor. C, announced in Cohen 1985, p. 194) See also at stable splitting of mapping spaces.

Here

• $\wedge$ denotes the smash product,

• $\mathbb{Z}/k$ a cyclic group,

• $B G$ the classifying space of a Lie group $G$ and $E G$ its universal principal bundle,

• $(-)_+$ the adjoining of a base point

• and $(-)\wedge_{G} (-) \;\coloneqq\; \big((-) \wedge (-)\big)/G$.

Example

For $X = S^n$ the n-sphere, we have $\Sigma X \,=\, S^{n+1}$ and hence

$$\Omega^\infty \Sigma^\infty \big( Maps(S^1, S^{n+1}) \sslash S^1 \big) \;\; \simeq \;\; \big( \Omega^\infty \Sigma^\infty B S^1 \big) \times \underset{ n \in \mathbb{N}_{\geq 1} }{\prod} \Omega^\infty \Sigma^\infty \big( (E (\mathbb{Z}/n))_+ \wedge_{\mathbb{Z}/n} (S^{n})^{\wedge^n} \big)$$

More on this case in Hingston 1992.

Rational homotopy

For the rational homotopy type of cyclic loop spaces see at Sullivan model for free loop space.

Related concepts

• double dimensional reduction

• string topology

• cyclic loop stack

• free loop space, free loop stack

References

The notion of the cyclic loop space of a topological space appears as:

• Ralph Cohen, p. 194 of: A Model for the Free Loop Space of a Suspension, in: Haynes Miller, Douglas Ravenel (eds.), Algebraic Topology, Lecture Notes in Mathematics 1286, Springer 1985 (doi:10.1007/BFb0078743)

• Gunnar Carlsson, Ralph Cohen, The cyclic groups and the free loop space, Commentarii Mathematici Helvetici 62 (1987) 423–449 (doi:10.1007/BF02564455, dml:140092)

• 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 (doi:10.1007/BFb0078045)

• David Chataur, A bordism approach to string topology, International Mathematics Research Notices 2005 46, (2005) (arXiv:math/0306080, doi:10.1155/IMRN.2005.2829)

• Marcel Bökstedt, Iver Ottosen, A spectral sequence for string cohomology, Topology Volume 44, Issue 6, November 2005, Pages 1181-1212 (arXiv:math/0411571, doi:10.1016/j.top.2005.04.006)

and in their relation to Hochschild homology/cyclic homology:

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

• 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)

and specifically for complex projective spaces in Loday (2015), Sec 4.

Discussion in the context of the transchromatic character:

• Nathaniel Stapleton, Transchromatic twisted character maps, J. Homotopy Relat. Struct. 10, 29–61 (2015). (arXiv:1304.5194, doi:10.1007/s40062-013-0040-9)

Specifically on cyclic loop spaces of n-spheres:

• Nancy Hingston, An Equivariant Model for the Free Loop Space of $S^N$, American Journal of Mathematics 114 1 (1992) 139-155 (doi:10.2307/2374740, jstor:2374740)

See also:

• Urs Frauenfelder, Dihedral homology and the moon, J. Fixed Point Theory Appl. 14 (2013) 55–69 (arXiv:1204.4549, doi:10.1007/s11784-013-0146-z)

A version of the cyclic loop stack of orbifolds, or at least its restriction to constant loops, namely Huan's inertia orbifold, is discussed in the context of equivariant elliptic cohomology via Tate K-theory in:

• Zhen Huan, Def. 2.14 of: Quasi-Elliptic Cohomology I, Advances in Mathematics, Volume 337, 15 October 2018, Pages 107-138 (arXiv:1805.06305, doi:10.1016/j.aim.2018.08.007)

following

• Zhen Huan, Section 2.1.2 of: Quasi-elliptic cohomology, 2017 (hdl)

and recalled/expanded on in several followup articles, such as in

• Zhen Huan, Section 2 of Quasi-theories (arXiv:1809.06651)

The above formulation of cyclic loop spaces, in the generality of ∞-stacks, as right base change to the delooping of the circle group, and its relation to double dimensional reduction in brane-physics, is due to:

• Vincent Braunack-Mayer, Hisham Sati, Urs Schreiber: Section 2.2 of Gauge enhancement of Super M-Branes via rational parameterized stable homotopy theory, Communications in Mathematical Physics, 371 197 (2019) (doi:10.1007/s00220-019-03441-4, arXiv:1806.01115)

following the analogous discussion in rational homotopy theory in

• Domenico Fiorenza, Hisham Sati, Urs Schreiber, Section 3 of: T-Duality from super Lie $n$-algebra cocycles for super $p$-branes, ATMP Volume 22 (2018) Number 5, doi:10.4310/ATMP.2018.v22.n5.a3, arXiv:1611.06536)

with exposition in

• Urs Schreiber, Section 4 of: Super Lie n-algebra of Super p-branes (2016)