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In 1985, S.~Halperin conjectured in the topological context of maximal free torus actions on topological manifolds, that:

If $X$ is a topological space, then $$\dim H^*(X;\mathbb Q)\geq 2^{rk(X)}.$$

Where $rk(X):=\max\{n\in\mathbb{N};\text{ such that }\mathbb{T}^n\text{ acts almost freely on } X\}$ is the toral rank of $X$.

My first question: TRC is obvious for $rk(X)=0$ or 1. Do we know for which values of $rk(X)\geq 2$ the conjecture holds?.

Any comments and references are welcome

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Volker Puppe has proved the conjecture for $rk_0(X)\le 3$, see the introduction here with a reference to the article of Puppe. There seems to be no general result for $rk(X)\ge 2$. The Lie algebra version, for nilpotent Lie algebras, has been proved for many classes of nilpotent Lie algebras (e.g., $2$-step nilpotent, or low dimension, see Proposition $2.2.7$ here).

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  • $\begingroup$ Thanks Dietrich. Just note that following both USTIONOVSKY introduction and Puppe Theorem 1.1, the rational TRC is resolved for $rk_0(X)\geq 3$. $\endgroup$
    – MyIsmail
    Commented Mar 27, 2015 at 20:47
  • $\begingroup$ Yes, thank you. Ustionovsky says $m\le 3$, right ? $\endgroup$
    – Dietrich Burde
    Commented Mar 27, 2015 at 21:05
  • $\begingroup$ Yes, TRC is ok for $rk_0(X)\leq 3$. For $rk_0(X)\geq 3$, Puppe bounded the rational cohomological dimension by 2(r+1). Is there any other best approximation? $\endgroup$
    – MyIsmail
    Commented Mar 28, 2015 at 8:03
  • $\begingroup$ I'v found the following: Aman showed in 2012 see arXiv:1204.6276 that $$\dim H^*(X;\mathbb Q)\geq 2(r+[r/3]).$$ 20 years before, Hilali proved in 1990 (see Theorem D) that $$\dim H^*(X;\mathbb Q)\geq 2(r^2+r).$$ $\endgroup$
    – MyIsmail
    Commented Mar 31, 2015 at 10:08
  • $\begingroup$ @MyIsmail: I think the $2$ in the second bound should be $1/2$. $\endgroup$
    – Mark Grant
    Commented Apr 3, 2015 at 17:10

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