nLab
iterated loop space

Contents

Context

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

A multiple loop space.

A grouplike E-k algebra in Top.

An iterated loop space object in Top.

Properties

Higher group structure

See at May recognition theorem.

Homotopy

Proposition

For AA a pointed homotopy type, hence an ∞-groupoid equipped with a base point *pt AA\ast \xrightarrow{ pt_A } A, then for nn \,\in\, \mathbb{N}, the n-fold loop space of AA is the homotopy fiber of the basepoint-evaluation map on the mapping space from the homotopy type of the n-sphere:

Ω nAhofib(ev *)Maps(ʃS n,A)ev *A \Omega^n A \xrightarrow{ hofib(ev_{\ast}) } Maps \big( ʃ S^n ,\, A \big) \xrightarrow{ ev_\ast } A

Proof

We may present the sequence in the classical model structure on topological spaces or the classical model structure on simplicial sets, in the latter case we may assume that AA is presented by a Kan complex, so that, in either case, it is a fibrant object.

In either case, the canonical model for the iterated loop space is evidently the ordinary 1-category-theoretic fiber of the evaluation map out of the internal hom:

Ω nAev *Maps(ʃS d,A)ev *A. \Omega^n A \xrightarrow{ ev_\ast } Maps( ʃ S^d ,\, A ) \xrightarrow{ ev_\ast } A \,.

Moreover, the evaluation map is equivalently the image of the point inclusion under the internal hom-functor

ev *=Maps(*S n,A). ev_\ast \;=\; Maps( \ast \to S^n ,\, A ) \,.

Since either model category is a cartesian closed monoidal model category, hence an enriched model category over itself (this Exp.) and since the canonical model for *ʃS n\ast \to ʃ S^n is a cofibration, in either case, the pullback-power axiom implies that ev *ev_\ast is a fibration. Therefore its ordinary fiber above models the homotopy fiber, and the claim follows.

Corollary

The homotopy groups of the mapping space Maps(ʃS n,A)Maps(ʃ S^n ,\, A) out of an n-sphere form a long exact sequence with those of AA, of the following form:

Proof

This is the long exact sequence of homotopy groups applied to the homotopy fiber sequence from Prop. .

Example

If AGrp A \,\in\, Grp_\infty is n-truncated, then the evaluation map out of the mapping space from the (n+2)-sphere into it is a weak homotopy equivalence:

τ n(A)AAAAAAAAAAAMaps(ʃS n+2,A)W whev *A. \tau_{n}(A) \,\simeq\, A {\phantom{AAAAA}} \Rightarrow {\phantom{AAAAA}} Maps \big( ʃ S^{n+2} ,\, A \big) \underoverset {\in \mathrm{W}_{wh}} {ev_\ast} {\longrightarrow} A \,.

Proof

By assumption, the long exact sequence from Cor. collapses to exact segments of the form

*π (Maps(ʃS n+2,A))π (ev *)π (A)*. \ast \to \pi_{\bullet} \big( Maps(ʃ S^{n+2},\, A) \big) \xrightarrow{ \;\;\; \pi_\bullet(ev_\ast) \;\;\; } \pi_\bullet(A) \to \ast \,.

Cohomology

Proposition

(rational cohomology of iterated loop space of the 2k-sphere)

Let

1D<n=2k 1 \leq D \lt n = 2k \in \mathbb{N}

(hence two positive natural numbers, one of them required to be even and the other required to be smaller than the first) and consider the D-fold loop space Ω DS n\Omega^D S^n of the n-sphere.

Its rational cohomology ring is the free graded-commutative algebra over \mathbb{Q} on one generator e nDe_{n-D} of degree nDn - D and one generator a 2nD1a_{2n - D - 1} of degree 2nD12n - D - 1:

H (Ω DS n,)[e nD,a 2nD1]. H^\bullet \big( \Omega^D S^n , \mathbb{Q} \big) \;\simeq\; \mathbb{Q}\big[ e_{n - D}, a_{2n - D - 1} \big] \,.

(Kallel-Sjerve 99, Prop. 4.10)


Homology

See at homology of iterated loop spaces.


Relation to configuration spaces of points

Proposition

(iterated loop spaces equivalent to configuration spaces of points)

For

  1. dd \in \mathbb{N}, d1d \geq 1 a natural number with d\mathbb{R}^d denoting the Cartesian space/Euclidean space of that dimension,

  2. YY a manifold, with non-empty boundary so that Y/YY / \partial Y is connected,

the electric field map/scanning map constitutes a homotopy equivalence

Conf( d,Y)scanΩ dΣ d(Y/Y) Conf\left( \mathbb{R}^d, Y \right) \overset{scan}{\longrightarrow} \Omega^d \Sigma^d (Y/\partial Y)

between

  1. the configuration space of arbitrary points in d×Y\mathbb{R}^d \times Y vanishing at the boundary (Def. )

  2. the d-fold loop space of the dd-fold reduced suspension of the quotient space Y/YY / \partial Y (regarded as a pointed topological space with basepoint [Y][\partial Y]).

In particular when Y=𝔻 kY = \mathbb{D}^k is the closed ball of dimension k1k \geq 1 this gives a homotopy equivalence

Conf( d,𝔻 k)scanΩ dS d+k Conf\left( \mathbb{R}^d, \mathbb{D}^k \right) \overset{scan}{\longrightarrow} \Omega^d S^{ d + k }

with the d-fold loop space of the (d+k)-sphere.

(May 72, Theorem 2.7, Segal 73, Theorem 3, see Bödigheimer 87, Example 13)

Proposition

(stable splitting of mapping spaces out of Euclidean space/n-spheres)

For

  1. dd \in \mathbb{N}, d1d \geq 1 a natural number with d\mathbb{R}^d denoting the Cartesian space/Euclidean space of that dimension,

  2. YY a manifold, with non-empty boundary so that Y/YY / \partial Y is connected,

there is a stable weak homotopy equivalence

Σ Conf( d,Y)nΣ Conf n( d,Y) \Sigma^\infty Conf(\mathbb{R}^d, Y) \overset{\simeq}{\longrightarrow} \underset{n \in \mathbb{N}}{\oplus} \Sigma^\infty Conf_n(\mathbb{R}^d, Y)

between

  1. the suspension spectrum of the configuration space of an arbitrary number of points in d×Y\mathbb{R}^d \times Y vanishing at the boundary and distinct already as points of d\mathbb{R}^d (Def. )

  2. the direct sum (hence: wedge sum) of suspension spectra of the configuration spaces of a fixed number of points in d×Y\mathbb{R}^d \times Y, vanishing at the boundary and distinct already as points in d\mathbb{R}^d (also Def. ).

Combined with the stabilization of the electric field map/scanning map homotopy equivalence from Prop. this yields a stable weak homotopy equivalence

Maps cp( d,Σ d(Y/Y))=Maps */(S d,Σ d(Y/Y))=Ω dΣ d(Y/Y)Σ scanΣ Conf( d,Y)nΣ Conf n( d,Y) Maps_{cp}(\mathbb{R}^d, \Sigma^d (Y / \partial Y)) = Maps^{\ast/}( S^d, \Sigma^d (Y / \partial Y)) = \Omega^d \Sigma^d (Y/\partial Y) \underoverset{\Sigma^\infty scan}{\simeq}{\longrightarrow} \Sigma^\infty Conf(\mathbb{R}^d, Y) \overset{\simeq}{\longrightarrow} \underset{n \in \mathbb{N}}{\oplus} \Sigma^\infty Conf_n(\mathbb{R}^d, Y)

between the latter direct sum and the suspension spectrum of the mapping space of pointed continuous functions from the d-sphere to the dd-fold reduced suspension of Y/YY / \partial Y.

(Snaith 74, theorem 1.1, Bödigheimer 87, Example 2)

In fact by Bödigheimer 87, Example 5 this equivalence still holds with YY treated on the same footing as d\mathbb{R}^d, hence with Conf n( d,Y)Conf_n(\mathbb{R}^d, Y) on the right replaced by Conf n( d×Y)Conf_n(\mathbb{R}^d \times Y) in the well-adjusted notation of Def. :

Maps cp( d,Σ d(Y/Y))=Maps */(S d,Σ d(Y/Y))nΣ Conf n( d×Y) Maps_{cp}(\mathbb{R}^d, \Sigma^d (Y / \partial Y)) = Maps^{\ast/}( S^d, \Sigma^d (Y / \partial Y)) \overset{\simeq}{\longrightarrow} \underset{n \in \mathbb{N}}{\oplus} \Sigma^\infty Conf_n(\mathbb{R}^d \times Y)
(∞,1)-operad∞-algebragrouplike versionin Topgenerally
A-∞ operadA-∞ algebra∞-groupA-∞ space, e.g. loop spaceloop space object
E-k operadE-k algebrak-monoidal ∞-groupiterated loop spaceiterated loop space object
E-∞ operadE-∞ algebraabelian ∞-groupE-∞ space, if grouplike: infinite loop space \simeq ∞-spaceinfinite loop space object
\simeq connective spectrum\simeq connective spectrum object
stabilizationspectrumspectrum object

References

General

Relation to configuration spaces of points

In relation to configuration spaces of points:

Rational cohomology

On ordinary cohomology of iterated loop spaces in relation to configuration spaces of points (see also at graph complex):

On the rational cohomology:

Last revised on October 1, 2021 at 05:34:14. See the history of this page for a list of all contributions to it.