nLab
loop space

Idea

A loop space is a loop space object in Top.

Definition

Let $\mathrm{Top}$ be a nice category of topological spaces, in particular one which is complete, cocomplete, and cartesian closed. Let $(S^1,\mathrm{pt})$ 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

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

in the category ${\mathrm{Top}}_{*}$ of based spaces. Explicitly, it is given by the pullback in $\mathrm{Top}$

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

(using exponentials to denote internal homs in $\mathrm{Top}$), 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 ${\mathrm{Top}}_{*}$ is symmetric monoidal closed; its monoidal product is called the smash product, often denoted $\wedge$. In particular, the loop space functor

$$\Omega =\mathrm{hom}((S^1,\mathrm{pt}),-):{\mathrm{Top}}_{*}\to {\mathrm{Top}}_{*}$$\Omega = hom((S^1, pt), -): Top_* \to Top_*

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

$$\begin{array}{cccc}& 1\times X+S^1\times 1& \to & 1\\ (\mathrm{pt}\times X,S^1\times *)& \downarrow & & \downarrow \\ & S^1\times X& \to & SX\end{array}$$\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.

Structure on loop spaces

A loop space is an example of a homotopy-associative space?, or H-space. In fact, loop spaces admit a 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.

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 $S⊣\Omega$ 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 the Stasheff operad? $K$.

Very stubby article; much work remains.

Local homotopy properties of loop spaces

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

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

but a much simpler and direct proof for general $X$ and $P=I$ or $P=S^1$ is possible.

David Roberts: The following claim is conjectural, but for $n=2$ it is in my thesis. $n=1$ has been known for ages - I don’t know where it was first, but it is in Ronnie Brown’s topology textbook.

Theorem

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.