I'm wondering if there is any elementary example of a space with precisely two non-trivial homotopy groups.
Let $X$ be a connected CW complex with precisely two non-trivial homotopy groups $\pi_p$ and $\pi_q$, with $p<q$, and a trivial $\pi_1$ action on $\pi_q$ if $p=1$.
In this situation, the homotopy type of $X$ is classified by a unique Postnikov invariant, which is a class in $H^{q+1}(B^{p}(\pi_p(X)),\pi_q(X))$.
The trivial cohomology classes correspond to products of Eilenberg-Mac Lane spaces.
Non-trivial classes give non-trivial examples.
For instance, if $\pi_1 = \pi_7 = \mathbb Z_2$, there is a non-trivial $X$ obtained by the non-trivial element of $H^8(B\mathbb Z_2,\mathbb Z_2) = H^8(\mathbb RP^\infty,\mathbb Z_2) = \mathbb Z_2$.
The space $X$ itself is obtained as the homotopy fiber of the corresponding map $\alpha:\mathbb RP^\infty\to K(\mathbb Z_2,8)$.
This is usually referred to as a "twisted product of Eilenberg–Mac Lane spaces", but even in this simple case it's a little obscure to me.
Is there any example with a name?
Best Answer
The issue is that as long as you're thinking about spaces familiar examples are things like manifolds or CW complexes and these are typically either aspherical or have homotopy groups in arbitrarily high degrees. There's sort of an uncertainty principle preventing spaces from having both simple homotopy and simple cohomology, and most familiar spaces are designed to have simple cohomology.
Instead the simplest examples come from higher category theory. We can construct spaces with nontrivial $\pi_1$ and $\pi_2$, for example, by constructing connected $2$-groupoids (then taking the geometric realization of their nerves, to be specific), or equivalently (after taking loop spaces) constructing $2$-groups (then taking their classifying spaces). An example of some practical interest is that every group $G$ has an automorphism $2$-group $\textbf{Aut}(G)$, whose objects are automorphisms of $G$ and whose morphisms are given by pointwise conjugation. It satisfies
$$\pi_0(\textbf{Aut}(G)) \cong \text{Out}(G), \pi_1(\textbf{Aut}(G)) \cong Z(G)$$
and it classifies arbitrary extensions in the sense that extensions $1 \to N \to G \to H \to 1$ are classified by maps $H \to \textbf{Aut}(N)$. Unfortunately I have no idea how to compute the Postnikov invariants of these things. In any case the classifying spaces $B \textbf{Aut}(N)$ furnish examples of spaces with only nontrivial $\pi_1$ and $\pi_2$.
The Postnikov invariant $k \in H^3(B\pi_1, \pi_2)$ of a connected $2$-groupoid can be understood more concretely in terms of the corresponding $2$-group as follows, at least in the case of trivial $\pi_1$ action. The $2$-group is a "groupal groupoid," namely a monoidal category $(M, \otimes)$ in which every object and every morphism is invertible. You can imagine trying to build such a thing by just taking $\pi_1$ to be the group of objects and $\pi_2$ to be the group of automorphisms of the unit object, which then must become the automorphisms of every object. The additional data you need to provide to define a monoidal category is the associator
$$\alpha(g, h, k) : (gh)k \cong g(hk)$$
which, since we've strictified the monoidal structure so $(gh)k = g(hk) = ghk$ on the nose, is an element of $\pi_2$ on the nose. Now if you write down what the axioms constraining the associator are in this case you get the condition satisfied by a $3$-cocycle on $\pi_1$ with coefficients in $\pi_2$. Moreover two associators give equivalent monoidal categories iff the corresponding $3$-cocycles are cohomologous. (There are some details to check here involving the unit but I don't think they substantially affect the story.)
There's a similar story you can tell about spaces with nontrivial $\pi_2$ and $\pi_3$ in terms of (again depending on the number of loops you want to take) simply connected $3$-groupoids, connected $3$-groups, or braided $2$-groups. Here, in terms of the braided $2$-group picture, I believe the Postnikov invariant $k \in H^4(B^2 \pi_2, \pi_3)$ corresponds to the braiding.