Contents

# Contents

## 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.

## References

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

and in their relation to Hochschild homology/cyclic homology:

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

Discussion in the context of the transchromatic character:

Specifically on cyclic loop spaces of n-spheres:

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:

following

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

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

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:

following the analogous discussion in rational homotopy theory in

with exposition in

Last revised on November 28, 2022 at 11:26:07. See the history of this page for a list of all contributions to it.