Contents

# Contents

## Idea

A multiple loop space.

A grouplike E-k algebra in Top.

## Properties

### Higher group structure

See at May recognition theorem.

### Homotopy

###### Proposition

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

$\Omega^n A \xrightarrow{ hofib(ev_{\ast}) } Maps \big( ʃ S^n ,\, A \big) \xrightarrow{ ev_\ast } A$

###### 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 $A$ 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:

$\Omega^n A \xrightarrow{ ev_\ast } Maps( ʃ S^d ,\, A ) \xrightarrow{ ev_\ast } A \,.$

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

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

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 $\ast \to ʃ S^n$ is a cofibration, in either case, the pullback-power axiom implies that $ev_\ast$ is a fibration. Therefore its ordinary fiber above models the homotopy fiber, and the claim follows.

###### Corollary

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

###### Proof

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

###### Example

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

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

###### Proof

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

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

### Cohomology

###### Proposition

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

Let

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

(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 $\Omega^D S^n$ of the n-sphere.

Its rational cohomology ring is the free graded-commutative algebra over $\mathbb{Q}$ on one generator $e_{n-D}$ of degree $n - D$ and one generator $a_{2n - D - 1}$ of degree $2n - D - 1$:

$H^\bullet \big( \Omega^D S^n , \mathbb{Q} \big) \;\simeq\; \mathbb{Q}\big[ e_{n - D}, a_{2n - D - 1} \big] \,.$

### Relation to configuration spaces of points

###### Proposition

(iterated loop spaces equivalent to configuration spaces of points)

For

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

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

the electric field map/scanning map constitutes a homotopy equivalence

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

between

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

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

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

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

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

###### Proposition

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

For

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

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

there is a stable weak homotopy equivalence

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

between

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

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

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

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

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

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

$Maps_{cp}(\mathbb{R}^d, \Sigma^d (Y / \partial Y)) = Maps^{\ast/}( S^d, \Sigma^d (Y / \partial Y)) \overset{\simeq}{\longrightarrow} \underset{n \in \mathbb{N}}{\oplus} \Sigma^\infty Conf_n(\mathbb{R}^d \times Y)$
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

### 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: