Suppose we have some pointed connected topological space $X$. How can we determine if there exists a space $BX$, called delooping of $X$, such that its space of based loops $\Omega BX$ is homotopy equivalent to $X$? More generally, when can we deloop a space $n$ times?
Abstract nonsense
There is a general theory answering this question, going back to J.D. Stasheff's 1963 papers "Homotopy Associativity of H-Spaces. I, II" and vastly developed later. The basic answer is that $X$ should carry an action of $A^\infty$ operad, corresponding to composition of loops. This composition is not associative (at least with an obvious definition), but associativity fails in a well-behaved way — up to coherent higher homotopies. If $X$ is an iterated loop space $X = \Omega^n Y$, then it carries $n$ compatible actions of $A^\infty$, because there are roughly $n$ really different ways to compose two $n$-dimensional spheres in $Y$: a choice of composition corresponds to a map $S^n \to S^n \vee S^n$, which can be obtain by contracting some base-pointed equator on $S^n$, and there are $n$ really different ways to choose it. Composition of $n$-dimensional spheres is also not associative, and different choices of composition do not commute, but all these statements fail up to coherent higher homotopies, encoded in an action of so-called $E_n$ operad on $X$ ($E_1 = A^\infty$). In this case $X$ can be delooped up to $n$ times. An example of explicit construction is given in "The Geometry of Iterated Loop Spaces" of J.P. May: there is a natural right action of $E_n$ operad on the $n$-th suspension functor $\Sigma^n$, so we can consider "derived tensor product" $\Sigma^n \otimes_{E_n} X$. It is a geometric realization of simplicial space $$\Sigma^n X \leftarrow \Sigma^n E_n X \leftleftarrows \Sigma^n E_n E_n X \leftleftarrows \dots$$
(last term shoul have 3, not 2 arrows, but how can you do this with MO TeX?)
The problems
It's all nice and neat, but how can you determine if a space can have an action of $E_n$? In all examples I'm familiar with this action is supplied with the definition of $X$, because $X$ is defined as an iterated loop space, or a group, or have a natural action of something that is close to $E_n$, like little disks in some manifold $E_M$, or is a representing object for some cohomology theory. Further problem is, even if we find some group structure on $X$, it doesn't necessarily mean that we have captured all structure. For example, a group can be noncommutative, but still admit higher homotopies, assembling into the action of $E_n$ or even $E_\infty$. A simple example is the infinite-dimensional projective unitary group $PU(H)$, where $H$ is some separable Hilbert space. Kuiper's theorem states that $U(H)$ is contractible. Its center is $U(1)$ and we have a fiber sequence
$$U(1) \to U(H) \to PU(H)$$
This means that $PU(H)$ is $K(\mathbb{Z},2)$ and an infinity-loop space. While it may be that these higher commutativity homotopies can be written down explicitly, it is not at all obvious that they exist in the first place. If $X$ is an infinity-loop space, we can at least hope that after several iterations of taking loops we will get some periodicity, or an obviously commutative group (naive, but is there much choice?). If $X$ is $n$-loop space for finite $n$, then even this hope is futile.
While I'm not expecting that there is some way, besides finding a certain $E_n$ action, to prove that $X$ is $n$-loop space (but I still would be glad to hear any different methods), I do hope that it is possible to prove that $X$ is not $n$-loop. Possibly there are some homotopic or homological invariants that would be obstructions to finding such a structure. However, I can only think of obvious ones: $\pi_1$ should be abelian. Perhaps some homotopical or homological operations should also vanish for $n$-loop spaces, but I don't know how to describe them.
An iterative check for the existence of $A^\infty$ structure can be performed based on Milnor's theorem. If $X$ has $A^\infty$ action, then it is homotopy equivalent to a topological group $G$, and we get "projective space bundles"
$$G \to G^{\ast n} \to B_n$$
Here $B_n$ is some space, and $G^{\ast n}$ is n-th iterated join. For $G=S^1$ or $G=S^3$ this gives standard Hopf bundles over complex and quaternionic projective spaces. A theorem of Adams asserts that such bundles do not exist for spheres of other dimensions (except for $\mathbb{O}P^1$ and $\mathbb{O}P^2$, $G=S^7$). However, his methods seem to rely on $G$ being a sphere, and I am not sure how to prove or disprove the existence of a fibration with fiber $X$ and total space $X^{\ast n}$ for a general space $X$.
Question
Are there different ways to prove that $X$ is $n$-loop space, besides finding an explicit $E_n$ action? More importantly, how can one prove that such action does not exist? What homological/homotopical invariants and operations can be used to disprove its existence, or prove that some known $E_n$ structure can not be lifted to a $E_{n+k}$ structure?
Best Answer
The general question is indeed hard, but in view of Anton's side question to Dylan about compact Lie groups, I feel compelled to advertise a beautiful old result of John Hubbuck that everyone should know: If $X$ is a connected finite CW complex and a homotopy commutative $H$-space, then $X$ is homotopy equivalent (as an $H$-space) to a torus. The reference is: J.R. Hubbuck. On homotopy commutative H-spaces. Topology 8 1969 119–126. Thus the original question trivializes unless $X$ is infinite dimensional.