CW-complex, Hausdorff space, second-countable space, sober space
connected space, locally connected space, contractible space, locally contractible space
hom-set, hom-object, internal hom, exponential object, derived hom-space
loop space object, free loop space object, derived 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 $X$: a space of loops in $X$. Here $X$ 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 $X$. 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 $(S^1, pt)$ 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
in the category $Top_*$ of based spaces. Explicitly, it is given by the pullback in $Top$
(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
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
with basepoint provided by the right vertical arrow.
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:
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):
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.
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.
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.
There is a Quillen equivalence
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.
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
See also the references at looping and delooping.