loop space



A loop space is a loop space object, typically regarded in Top (in the context of topology) or in Ho(Top) or rather in ∞Grpd (in the context of homotopy theory).

As a topological space it is equivalently a mapping space from the circle to some pointed topological space XX: a space of loops in XX. Here XX and the loops might be equipped with further geometric structure such as smooth structure and then one may consider a smooth loop space, etc.

Strictly speaking, and as considered here, a loop space consists only of loops that start and end at a fixed base point in XX. Without this restriction one speaks of a free loop space.


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

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

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

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

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

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

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

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

1×X+S 1×1 1 (pt×X,S 1×*) S 1×X SX\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.


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 1S^1 carries a correspondingly rich co-algebraic structure, starting from the fact that the based space S 1S^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: XΩYX \simeq \Omega Y. In this circumstance, one calls YY a delooping of XX; an important example is where XX carries a topological group structure GG, and YY is the classifying space of GG.

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

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

which follows straight from the adjunction SΩS \dashv \Omega plus the fact that the suspension of S nS^n is S n+1S^{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):


An H-space XX admits a delooping if and only if the monoid π 0(X)\pi_0(X) induced from the H-space structure is a group, and the H-space XX structure can be extended to a structure of algebra over an operad over Stasheff’s A-∞ operad KK.

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 XX be locally 0-connected and semi-locally 1-connected? (i.e. it admits a universal covering space). The loop space ΩX\Omega X for any basepoint is locally path connected, as is the free loop space X S 1X^{S^1}. If XX is locally 1-connected and admits a basis of open sets UU such that π 2(U)π 2(X)\pi_2(U) \to \pi_2(X) is the zero map, then ΩX\Omega X is locally 0-connected and semi-locally 1-connected, and so admits a universal covering space.

In general, if XX is locally nn-connected, ΩX\Omega X is locally (n1)(n-1)-connected. This process can obviously be iterated up to nn times, so that Ω nX\Omega^n X is locally 0-connected. This can be weakened to locally (n1)(n-1)-connected and semi-locally nn-connected: this is just like the n=1n=1 case but replacing π 1\pi_1 with π n\pi_n (as was done in the previous paragraph with π 2\pi_2). We will actually define a space to be semi-locally nn-connected to include the condition that it is locally (n1)(n-1)-connected. This result was proved for more general mapping spaces X PX^P and various subspaces when XX is Hausdorff and PP a finite polyhedron in (Wada) but a much simpler and direct proof for general XX and P=IP = I or P=S 1P= S^1 is possible.


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

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


There is a Quillen equivalence

(GW¯):sGrpW¯ΩsSet 0 (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 GG, 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.

(∞,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 Γ-spaceinfinite loop space object
\simeq connective spectrum\simeq connective spectrum object
stabilizationspectrumspectrum object


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

  • Jim Stasheff, Homotopy associative HH-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

See also the references at looping and delooping.

Revised on February 22, 2017 02:53:15 by Urs Schreiber (