TITLE: Are there pairs of highly connected finite CW-complexes with the same homotopy groups?
QUESTION [36 upvotes]: Fix an integer n.  Can you find two finite CW-complexes X and Y which 
* are both n connected,
* are not homotopy equivalent, yet
* $\pi_q X \approx \pi_q Y$ for all $q$.
In Are there two non-homotopy equivalent spaces with equal homotopy groups? some solutions are given with n=0 or 1.  Along the same lines, you can get an example with n=3, as follows.  If $F\to E\to B$ is a fiber bundle of connected spaces such that the inclusion $F\to E$ is null homotopic, then there is a weak equivalence $\Omega B\approx F\times \Omega E$.  Thus two such fibrations with the same $F$ and $E$ have base spaces with isomorphic homotopy groups.
Let $E=S^{4m-1}\times S^{4n-1}$.  Think of the spheres as unit spheres in the quaternionic vector spaces $\mathbb{H}^m, \mathbb{H}^n$, so that the group of unit quarternions $S^3\subset \mathbb{H}$ acts freely on both.  Quotienting out by the action on one factor or another, we get fibrations
$$ S^3 \to E \to \mathbb{HP}^{m-1} \times S^{4n-1},\qquad S^3\to E\to S^{4m-1}\times \mathbb{HP}^{n-1}.$$
The inclusions of the fibers are null homotopic if $m,n>1$, so the base spaces have the same homotopy groups and are 3-connected, but aren't homotopy equivalent if $m\neq n$.
There aren't any n-connected lie groups (or even finite loop spaces) for $n\geq 3$, so you can't push this trick any further.  
Is there any way to approach this problem?  Or reduce it to some well-known hard problem?
(Note: the finiteness condition is crucial; without it, you can easily build examples using fibrations of Eilenberg-MacLane spaces, for instance.)

REPLY [3 votes]: There are quite a few finite CW-complexes $X$ and $Y$ that have homotopy equivalent loop spaces, thus have isomorphic homotopy groups, but themselves are not homotopy equivalent. Examples include taking $X=S^n\times S^n$ and $Y$ any simply connected space that has isomorphic cohomology ring to $X$, $X$ and $Y$ to be homotopically distinct closed simply connected $4$-manifolds with the same second Betti numbers, or more generally $(n-1)$-connected $2n$-manifolds at least when $n\neq 4,8$. This can be proved by obtaining homotopy decompositions of their loop spaces, showing the factors in the decomposititions of $\Omega X$ and $\Omega Y$ are the same.