The toral rank conjecture

The Toral rank conjecture (TRC), formulated in 1985 by Steve Halperin, states that whenever a compact torus \(T^r\) acts almost freely, i.e. only with finite stabilizers, on a compact Hausdorff space X, the inequality \(\dim H^*(X;Q) \geq \dim H^*(T^r;Q) = 2r\) holds. It is one of the most prominent open questions in the theory of topological transformation groups. We have discussed and summarized various results of the last 30 years on TRC.

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