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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?

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up vote 12 down vote accepted

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" by Blanc-Dwyer-Goerss studies this problem and how one might go about classifying realizations.

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