Let $X$ be a compact connected smooth manifold and $T$ a compact torus acting smoothly on $X$ without fixed points. What, in general, can be said about the topology of $X$ (ex. rational (co)homology)? I would appreciate any and all references and suggestions.
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The toral rank conjecture (or HalperinCarlsson conjecture) states that if $T^n$ acts with finite isotropy groups on the simplyconnected closed manifold $X$, then $$ \sum_i \dim H^i(X;\mathbb{Q})\ge 2^n. $$ There is an analogous statement in characteristic $p$, when $T^n$ acts freely. This is not yet proved, despite overwhelming evidence. You're sure to find relevant information in the many papers on this topic. 

