How to construct a CW complex $X$ with prescribed homotopy groups $\pi\_i(X)$ and prescribed actions of $\pi\_1(X)$ on the $\pi\_i(X)$'s?



Construct a CW complex with prescribed homotopy groups and actions of $\pi_1$.
One method is as follows.


1. Construct $Y\_i = K(\pi\_i, i)$ for $i \geq 2$ as a based space with an action of the group $\pi\_1$. You can do this manually (by attaching free orbits of cells along the group action) or canonically (using a functorial construction of Eilenberg-Mac Lane spaces, such as one obtained from the Dold-Kan correspondence and geometric realization).
2. Take $Y = \prod\_{i \geq 2} Y\_i$ as a based space with the correct "top" homotopy groups and an action of $\pi\_1$ realizing the desired action on the higher homotopy groups.
3. Take the Borel construction $X = E\pi\_1 \times\_{\pi\_1} Y$. This space, by construction, has fundamental group $\pi\_1$, has $E \pi\_1 \times Y \simeq Y$ as universal cover, and has the group of deck transformations given by the action of $\pi\_1$ on $E\pi\_1 \times Y$.


Of course, you don't *have* to take $Y$ to be a product of Eilenberg-Mac Lane spaces. Passing between universal covers and Borel constructions reduced your original question to finding a simply-connected space with the given higher homotopy groups and an action of $\pi\_1$ on the whole space.


You might also specify more structure; for example, the higher homotopy groups in the outlined construction are not related by Whitehead products or anything similar. This makes the problem harder. If you specify *all* the structure that the homotopy groups naturally have, you have a so-called $\Pi$-algebra; the paper "[The realization space of a $\Pi$-algebra](http://hopf.math.purdue.edu/cgi-bin/generate?/Blanc-Dwyer-Goerss/moduli)" by Blanc-Dwyer-Goerss studies this problem and how one might go about classifying realizations.


