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Let $X$ be a topological space. Suppose $\pi_1(X)$ is solvable, can we say something about $X$ ?

This question is probably broad, however I am interested in knowing if there is anything at all that can be said about such a space (or perhaps its universal cover, if it exists)

Thank you.

R_D
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  • Since solvable groups are nilpotent, and a nice chunk of algebraic topology requires you restrict to nilpotent fundamental group, it says you can do a nice chunk of algebraic topology. I don't know what else to say. –  Jul 25 '16 at 07:09
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    Solvable groups are nilpotent? Isn't it the other way around? See here – R_D Jul 25 '16 at 10:17
  • @MikeMiller, Could you give me an example of or a reference to a book/article(s) that deal with this nice chunk of algebraic topology. Thank you. – R_D Jul 25 '16 at 10:25
  • Oops, I guess I was falling asleep when I wrote that. Sorry! (I'm thinking, off the top of my head, of rational homotopy theory, but I remember this assumption appearing elsewhere.) –  Jul 25 '16 at 14:27

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