3
$\begingroup$

Let $M$ be a connected, simply connected topological space, and $\pi_2(M)$ its second homotopy group. I am searching for some work/references/papers/... done about such a space with $\pi_2(M)$ which is non-finitely-generated.

Are there some known and interesting results?

Since the question is very broad, one can take an example: how does a space $M$ with $\pi_2(M)=\mathbb{Q}$ look like?

Thanks ;-)

Improve this question
$\endgroup$
4
  • 7
    $\begingroup$ This is way too broad... what exactly are you hoping to find out? $\endgroup$
    – Dylan Wilson
    Commented Aug 12, 2017 at 14:56
  • $\begingroup$ Yes, I know this is wide... I am a theoretical-mathematical physicist so for me usually M is more precisely a 4-dimensional smooth manifold, and in my current work I am investigating the case of a non integrable Lie Algebroid over M, and in a special case that implies that the second homotopy group of M has to be non-finitely-generated, and I am just wondering how does this kind of space look like?... What kind of topological consequences that can produce on $M$ ? ... I know, this is not really less broad... $\endgroup$
    – JeremyA
    Commented Aug 12, 2017 at 15:24
  • $\begingroup$ So you're looking for an $K(\Bbb Q, 2)$? $\endgroup$
    – Ali Caglayan
    Commented Aug 13, 2017 at 19:53
  • $\begingroup$ Thanks for your answer, but sorry, I don't know what is an $K(\mathbb{Q},2)$ ... Could you describe it to me? Or give me the name of it so I could search by my own? Thanks very much $\endgroup$
    – JeremyA
    Commented Aug 13, 2017 at 20:32

0

Reset to default