nLab
iterated loop space

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 $A$ 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 $2n - D - 1$:

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

(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. $d \in \mathbb{N}$, $d \geq 1$ 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.

(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. $d \in \mathbb{N}$, $d \geq 1$ 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 $Y / \partial Y$.

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

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. :

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

Related concepts

	(∞,1)-operad	∞-algebra	grouplike version	in Top	generally
	A-∞ operad	A-∞ algebra	∞-group	A-∞ space, e.g. loop space	loop space object
	E-k operad	E-k algebra	k-monoidal ∞-group	iterated loop space	iterated loop space object
	E-∞ operad	E-∞ algebra	abelian ∞-group	E-∞ space, if grouplike: infinite loop space $\simeq$ ∞-space	infinite loop space object
				$\simeq$ connective spectrum	$\simeq$ connective spectrum object
stabilization				spectrum	spectrum object

• looping and delooping, stabilization hypothesis

References

General

• Peter May, Infinite loop space theory, Bull. Amer. Math. Soc. Volume 83, Number 4 (1977), 456-494. (euclid:bams/1183538891)

  Infinite loop space theory revisited (pdf)

• John Frank Adams, Infinite loop spaces, Hermann Weyl lectures at IAS, Princeton University Press (1978) (ISBN:9780691082066, doi:10.1515/9781400821259)

• Peter May, The uniqueness of infinite loop space machines, Topology, vol 17, pp. 205-224 (1978) (pdf)

• Jacob Lurie, Section 5.1.3 of Higher Algebra

Relation to configuration spaces of points

In relation to configuration spaces of points:

• Peter May, The geometry of iterated loop spaces, Springer 1972 (pdf, doi:10.1007/BFb0067491)

• Graeme Segal, Configuration-spaces and iterated loop-spaces, Invent. Math. 21 (1973), 213–221. MR 0331377 (pdf)

• Victor Snaith, A stable decomposition of $\Omega^n S^n X$, Journal of the London Mathematical Society 7 (1974), 577 - 583 (pdf)

• Dusa McDuff, Configuration spaces of positive and negative particles, Topology Volume 14, Issue 1, March 1975, Pages 91-107 (doi:10.1016/0040-9383(75)90038-5)

• Carl-Friedrich Bödigheimer, Stable splittings of mapping spaces, Algebraic topology. Springer 1987. 174-187 (pdf, pdf)

Rational cohomology

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

• Carl-Friedrich Bödigheimer, Fred Cohen, L. Taylor, On the homology of configuration spaces, Topology Vol. 28 No. 1, p. 111-123 1989 (pdf)

On the rational cohomology:

• Sadok Kallel, Denis Sjerve, On Brace Products and the Structur eof Fibrations with Section, 1999 (pdf, pdf)