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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 know one can show $H_{T}^*(X)$ to be a torsion module over the eqivariant cohomology of a point, but perhaps there are more sweeping statements one can make. I would appreciate any and all references and suggestions.

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.

Let $X$ be a compact connected smooth manifold and $T$ a complex torus acting smoothly on $X$ without fixed points. What, in general, can be said about the topology of $X$ (ex. rational (co-)homology)? I know one can show $H_{T}^*(X)$ to be a torsion module over the eqivariant cohomology of a point, but perhaps there are more sweeping statements one can make. I would appreciate any and all references and suggestions.

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 know one can show $H_{T}^*(X)$ to be a torsion module over the eqivariant cohomology of a point, but perhaps there are more sweeping statements one can make. I would appreciate any and all references and suggestions.

Let $X$ be a compact smooth manifold and $T$ a complex torus acting smoothly on $X$ without fixed points. What, in general, can be said about the topology of $X$ (ex. rational (co-)homology)? I know one can show $H_{T}^*(X)$ to be a torsion module over the eqivariant cohomology of a point, but perhaps there are more sweeping statements one can make. I would appreciate any and all references and suggestions.

Let $X$ be a compact connected smooth manifold and $T$ a complex torus acting smoothly on $X$ without fixed points. What, in general, can be said about the topology of $X$ (ex. rational (co-)homology)? I know one can show $H_{T}^*(X)$ to be a torsion module over the eqivariant cohomology of a point, but perhaps there are more sweeping statements one can make. I would appreciate any and all references and suggestions.