nLab loop space

Idea

In the strict sense of the word a loop space in topology for a given pointed topological space $X$ is the mapping space (with its compact-open topology, see the example there) $Maps_\ast(S^1, X)$ of continuous functions from the circle to $X$, such that they take the given basepoint of the circle to the prescribed basepoint in $X$ (or if one drops this condition, then one speaks of the free loop space). One such continuous function may be thought of as a continuous loop in $X$, and hence the mapping space is the space of all these loops.

If here $X$ is equipped with further structure, such as smooth structure (e.g. a smooth manifold), then one may in good cases find such extra structure also on the loop space, for instance to form a smooth loop space, etc. See at manifolds of mapping spaces for more on this.

If one regards this construction not in point-set topology but in classical homotopy theory of topological spaces (equivalently ∞Grpd), then, up to weak homotopy equivalence, the loop space is equivalent to the homotopy fiber product of the basepoint inclusion $\ast \to X$ along itself.

Definition

Let Top be a nice category of topological spaces, in particular one which is complete, cocomplete, and cartesian closed. Let $(S^1, pt)$ be the circle, i.e., 1-dimensional sphere, with chosen basepoint, and let $(X, *)$ be a space with a chosen basepoint. Then the loop space of $X$ (at $*$) is an internal hom

$$\Omega X = hom((S^1, pt), (X, *))$$

in the category $Top_*$ of based spaces. Explicitly, it is given by the pullback in $Top$

$$\array{ \Omega X & \to & 1\\ \downarrow & & \downarrow *\\ X^{S^1} & \underset{X^{pt}}{\to} & X^1 }$$

(using exponentials to denote internal homs in $Top$), in other words the function space of basepoint-preserving maps $S^1 \to X$, whose basepoint is the constant map $S^1 \to X$ at the basepoint of $X$.

The category $Top_*$ is symmetric monoidal closed; its monoidal product is called the smash product, often denoted $\wedge$. In particular, the loop space functor

$$\Omega = hom((S^1, pt), -): Top_* \to Top_*$$

has a left adjoint obtained by taking smash product with $(S^1, pt)$. This left adjoint $S: Top_* \to Top_*$ is called the suspension functor. Explicitly, the suspension $S X$ is formed as the pushout

$$\array{ & 1 \times X + S^1 \times 1& \to & 1\\ (pt \times X, S^1 \times *) & \downarrow & & \downarrow \\ & S^1 \times X & \to & S X }$$

with basepoint provided by the right vertical arrow.

Properties

Homotopy-associative structure

A loop space is an example of a A-∞ space, in particular it is an H-space. Loop spaces admit this rich algebraic structure which arises from the fact that the based space $S^1$ carries a correspondingly rich co-algebraic structure, starting from the fact that the based space $S^1$ is an H-cogroup.

The description of this structure on loop spaces has been the very motivation for the introduction of the notion of operad and algebra over an operad in (May).

An important theoretical consideration is when an H-space, and particularly one having the type of a CW-complex, has the homotopy type of a loop space of another CW-complex: $X \simeq \Omega Y$. In this circumstance, one calls $Y$ a delooping of $X$; an important example is where $X$ carries a topological group structure $G$, and $Y$ is the classifying space of $G$.

The most basic fact about deloopings is the shift in homotopy groups:

• $\pi_n(\Omega Y) \cong \pi_{n+1}(Y)$

which follows straight from the adjunction $S \dashv \Omega$ plus the fact that the suspension of $S^n$ is $S^{n+1}$. (This isomorphism needs to be developed at greater length.)

The modern study of the question “when can an H-space be delooped?” was inaugurated by Jim Stasheff. The basic theorem is as follows (all spaces assumed to be CW-complexes):

Theorem

An H-space $X$ admits a delooping if and only if the monoid $\pi_0(X)$ induced from the H-space structure is a group, and the H-space $X$ structure can be extended to a structure of algebra over an operad over Stasheff‘s A-∞ operad $K$.

This is due to (Stasheff). The analogous statement holds true in every (∞,1)-topos other than Top. Details on this more general statement are at loop space object and at groupoid object in an (∞,1)-category.

Local homotopy properties

Let the space $X$ be locally 0-connected and semi-locally 1-connected (i.e. it admits a universal covering space). The loop space $\Omega X$ for any basepoint is locally path connected, as is the free loop space $X^{S^1}$. If $X$ is locally 1-connected and admits a basis of open sets $U$ such that $\pi_2(U) \to \pi_2(X)$ is the zero map, then $\Omega X$ is locally 0-connected and semi-locally 1-connected, and so admits a universal covering space.

In general, if $X$ is locally $n$-connected, $\Omega X$ is locally $(n-1)$-connected. This process can obviously be iterated up to $n$ times, so that $\Omega^n X$ is locally 0-connected. This can be weakened to locally $(n-1)$-connected and semi-locally $n$-connected: this is just like the $n=1$ case but replacing $\pi_1$ with $\pi_n$ (as was done in the previous paragraph with $\pi_2$). We will actually define a space to be semi-locally $n$-connected to include the condition that it is locally $(n-1)$-connected. This result was proved for more general mapping spaces $X^P$ and various subspaces when $X$ is Hausdorff and $P$ a finite polyhedron in (Wada) but a much simpler and direct proof for general $X$ and $P = I$ or $P= S^1$ is possible.

Conjecture

The fundamental $n$-groupoid of a space $X$ (Trimblean for choice) can be topologised to be an internal $n$-groupoid in $\Top$ when $X$ is semi-locally $n$-connected. Furthermore, the homotopy groups of the $n$-groupoid, a priori topological groups, are discrete.

For $n=2$, this is in David Roberts's thesis. For $n=1$, it has been known for ages and is in Ronnie Brown's topology textbook.

Homology of loop spaces

See at homology of loop spaces.

Models

There is a Quillen equivalence

$$(G \dashv \bar W) \;\colon\; sGrp \stackrel{\overset{\Omega}{\leftarrow}}{\underset{\bar W}{\to}} sSet_0$$

between the model structure on simplicial groups and the model structure on reduced simplicial sets, thus exhibiting both of these as models for infinity-groups (Kan 58). Its left adjoint $G$, the simplicial loop space construction, is a concrete model for the loop space construction with values in simplicial groups.

See also simplicial group and groupoid object in an (∞,1)-category for more details.

Related concepts

• path space

• loop space object, free loop space object

  loop space type

  • Sullivan model of loop space

  • delooping, looping and delooping

  • loop space, free loop space,

  • relative loop space

  • iterated loop space

  • infinite loop space

  • formal loop space

  • derived loop space

  • smooth loop space

  • loop space of a wedge of circles

• caloron correspondence

• suspension object

  • suspension

	(∞,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

• A-infinity space, model structure for dendroidal Cartesian fibrations

References

General

• Daniel Kan, On homotopy theory and c.s.s. groups, Ann. of Math. 68 (1958), 38-53

• Jim Stasheff, Homotopy associative $H$-spaces I, II, Trans. Amer. Math. Soc. 108, 1963, 275-312

• Peter May, The geometry of iterated loop spaces Lecture Notes in Mathematics 271 (1970) (pdf)

• H. Wada, Local connectivity of mapping spaces, Duke Mathematical Journal, vol ? (1955) pp 419-425

The simplicial loop group functor is discussed in chapter V, section 5 of

• Paul Goerss, Rick Jardine, Simplicial homotopy theory (web)

See also the references at looping and delooping.

Examples

On loop spaces of configuration spaces of points:

• Edward Fadell, Sufian Husseini, The space of loops on configuration spaces and the Majer-Terracini index, Topol. Methods Nonlinear Anal. Volume 11, Number 2 (1998), 249-271 (euclid:tmna/1476842829)

• Fred Cohen, Samuel Gitler, Loop spaces of configuration spaces, braid-like groups, and knots, In: Aguadé J., Broto C., Carles Casacuberta (eds.) Cohomological Methods in Homotopy Theory. Progress in Mathematics, vol 196. Birkhäuser, Basel (doi:10.1007/978-3-0348-8312-2_7)

• Toshitake Kohno, Loop spaces of configuration spaces and finite type invariants, Geom. Topol. Monogr. 4 (2002) 143-160 (arXiv:math/0211056)

• Fred Cohen, Samuel Gitler, On loop spaces of configuration spaces, Trans. Amer. Math. Soc. 354 (2002), no. 5, 1705–1748, (jstor:2693715, MR2002m:55020)

  (on ordinary homology of loop spaces of configuration spaces)