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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asked Jul 14, 2013 at 19:10
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1$\begingroup$ What type of result are you looking for? Obviously for every compact, connected, smooth manifold $Q$, taking $X=Q\times T$ with the obvious $T$-action, then $H^*_T(X)$ equals $H^*(Q)$. So, in this sense, there are no further restrictions on $H^*_T(X)$. $\endgroup$– Jason StarrCommented Jul 15, 2013 at 1:35
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$\begingroup$ Yes, that's quite right. The cohomology of any manifold can be realized as the equivariant cohomology of a $T$-manifold on which $T$ acts freely. However, I am not particularly interested in $H_T^*(X)$, but in $H^*(X)$. I will amend my question accordingly. $\endgroup$– Peter CrooksCommented Jul 15, 2013 at 2:11
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$\begingroup$ I would like to investigate non-equivariant ways to describe the topology of $X$. Anything about the cohomology of $X$ (or how I might find it) would be welcome, for instance. $\endgroup$– Peter CrooksCommented Jul 15, 2013 at 2:18
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The toral rank conjecture (or Halperin--Carlsson conjecture) states that if $T^n$ acts with finite isotropy groups on the simply-connected 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.
answered Jul 15, 2013 at 6:17