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Let $F$ be a (nontrivial) topological space that satisfies the following conditions: 1) $\pi_n(F)$ has a trivial action of $\pi_1(F)$ for $n>0$ and 2) its homology groups are finitely generated. Then $\pi_n(F)$ is finitely generated for $n>0$; it therefore makes sense to consider the following sum: $$I(F)=\sum_{q=1}\frac{\operatorname{Rank}(\pi_q(F))}{q}.$$ I have two questions:

When does $I(F)$ converge? When is $I(F)$ a natural number? (Like is there a (necessary and) sufficient condition for $I(F)$ to converge and/or be a natural number?)

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    $\begingroup$ For the second question, you have $S^1$, for example. It is easy to construct more examples. $\endgroup$
    – Espen Nielsen
    Commented May 25, 2015 at 18:51
  • $\begingroup$ @Espen Yeah, I did have that as an example. I hadn't phrased my question properly. It's fixed now! $\endgroup$
    – user62675
    Commented May 25, 2015 at 18:59
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    $\begingroup$ Finite generation is a condition not a consequence of your other assumption... Edit: Ok now it is. $\endgroup$
    – Dylan Wilson
    Commented May 25, 2015 at 19:00
  • $\begingroup$ @Dylan Fixed (I forgot (again) to say that its homology groups are finitely generated, which coupled with the other assumption says that the homotopy groups are finite (it's some result of Serre, I think)). Edit: see Theorem 1.7 here. $\endgroup$
    – user62675
    Commented May 25, 2015 at 19:03
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    $\begingroup$ If $F$ is simply connected and a finite CW-complex, then $I(F)<\infty$ iff $F$ has only finitely many nontrivial rational homotopy groups (see Theorem 2.33 here, for instance; I'm not sure whether the simply connected hypothesis is really necessary). If you don't demand $F$ to be a finite CW-complex, the question seems hopeless in full generality (for instance, $F$ could be an arbitrary product of spheres of different dimensions). Do you have any particular motivation for looking at $I(F)$? $\endgroup$
    – Eric Wofsey
    Commented May 25, 2015 at 19:26

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Suppose $F$ is a simply connected finite CW complex. Then it's known that exactly one of the following two things is true:

  • $F$ is rationally elliptic: its rational homotopy groups are finitely generated. In this case your sum clearly converges because it has finitely many terms.
  • $F$ is rationally hyperbolic: the ranks of its rational homotopy groups grow, on average, exponentially. In this case your sum clearly diverges because its terms don't go to zero.

For a simple example of the second case you can take $F = S^2 \vee S^2$.

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  • $\begingroup$ Thanks! Do you know when $I(F)$ can be a natural number? $\endgroup$
    – user62675
    Commented May 25, 2015 at 19:37
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    $\begingroup$ @Sanath: that doesn't seem like a particularly natural question to ask. Do you have any reason to expect that it is? (You can arrange for this sum to be a natural number by taking $F$ to be a suitable product of odd-dimensional spheres, for example. But I don't see the significance of this.) $\endgroup$
    – Qiaochu Yuan
    Commented May 25, 2015 at 19:38
  • $\begingroup$ True, it isn't a natural question to ask. But if $F=S^1$, then $I(F)$ is $1$. I was wondering if there were any other examples like that, and if so, could they be classified. $\endgroup$
    – user62675
    Commented May 25, 2015 at 19:41
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    $\begingroup$ You can consider Eilenberg MacLane Spaces: $K(\mathbb{Z}^{\oplus n},n)$. $\endgroup$
    – David C
    Commented May 25, 2015 at 19:45
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Maybe Félix, Halperin and Thomas recent work on ranks of homotopy groups of finite $1$-connected CW-complexes can be of interest regarding your questions:

"Exponential growth and an asymptotic formula for the ranks of homotopy groups of a finite 1-connected complex", Annals of Mathematics, 170 (2009), 443–464

http://annals.math.princeton.edu/wp-content/uploads/annals-v170-n1-p13-p.pdf

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  • $\begingroup$ David, correct me if I'm mistaken, but this paper considers the sum of the ranks, i.e., $\sum\mathrm{Rank}(\pi_q(F))$, not $\sum\frac{\mathrm{Rank}(\pi_q(F))}{q}$. (The denominator in the expression is like a "normalization".) $\endgroup$
    – user62675
    Commented May 25, 2015 at 19:18
  • $\begingroup$ You are right this paper considers sums of the form $$\sum_{q=k+2}^{k+n} Rank(\pi_q(X)).$$ This is related to the elliptic/hyperbolic dichotomy in homotopy theory. Your sum is convergent in the elliptic case. $\endgroup$
    – David C
    Commented May 25, 2015 at 19:25

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