nLab
cyclic loop space

Contents

Context

Topology

topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

Homotopy theory

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology

Introductions

Definitions

Paths and cylinders

Homotopy groups

Basic facts

Theorems

Contents

Idea

Any free loop space X\mathcal{L}X has a canonical action (infinity-action) of the circle group S 1S^1. The homotopy quotient (X)/S 1\mathcal{L}(X)/S^1 of this action might be called the cyclic loop space of XX, and is also known as the string space of XX (Chataur 2005. 4.8.1, Bökstedt & Ottosen 05, p. 1).

If X=Spec(A)X = Spec(A) is an affine variety regarded in derived algebraic geometry, then 𝒪(Spec(A))\mathcal{O}(\mathcal{L}Spec(A)) is the Hochschild homology of AA and 𝒪((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 (X//G)\mathcal{L}(X//G) is an BB \mathbb{Z}-action, then the restriction of the cyclic loop space to the constant loops constY//G(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 *BS 1\ast \to \mathbf{B} S^1

The cyclic loop space XS 1\mathcal{L}X \sslash S^1 is equivalently the right base change/dependent product along the canonical point inclusion *BS 1\ast \to B S^1 (this prop.) into the delooping of S 1S^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 XS 1\mathcal{L}X \sslash S^1 as cyclic cohomology 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 XS 1\mathcal{L}X \sslash S^1 is the cyclic homology HC HC_\bullet of the singular chains:

H (XS 1)HC (C (X)) 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 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 (XS 1)HC (Ω (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 XTopSpaces */X \in TopSpaces^{\ast/} a pointed topological space with reduced suspension denoted ΣX\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:

Ω Σ (Maps(S 1,ΣX)S 1)(Ω Σ BS 1)×k +Ω Σ ((E(/k)) + /kX n) \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:

Σ (Maps(S 1,ΣX) S 1(ES 1) +ES 1× S 1Maps(S 1,ΣX)ES 1× S 1*=BS 1)k 1Σ ((E/k) + /kX k) \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

Example

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

Ω Σ (Maps(S 1,S n+1)S 1)(Ω Σ BS 1)×n 1Ω Σ ((E(/n)) + /n(S n) n) \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:

Specifically on cyclic loop spaces of n-spheres:

See also:

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 August 18, 2021 at 11:00:50. See the history of this page for a list of all contributions to it.