Let X be a finite-dimensional topological space. An action of the torus T^r on X is almost free if all its isotropy subgroups are finite. Write operatornamehrk(X)=∑_i dim H^i(X). Toral rank conjecture. If T^r acts almost freely on X, then operatornamehrk(X)≥ 2^r. This is a central conjecture in transformation groups and equivariant topology, relating the rank of an almost-free torus action to the total Betti number of the space. Its resolution is not established by the supplied source, so it remains open here.

The toral rank conjecture

About 17 years old · traced to 2009

Let $X$ be a finite-dimensional topological space. An action of the torus $T^r$ on $X$ is almost free if all its isotropy subgroups are finite. Write

\operatorname{hrk}(X)=\sum_i \dim H^i(X).

Toral rank conjecture. If $T^r$ acts almost freely on $X$ , then

\operatorname{hrk}(X)\geq 2^r.

This is a central conjecture in transformation groups and equivariant topology, relating the rank of an almost-free torus action to the total Betti number of the space. Its resolution is not established by the supplied source, so it remains open here.

Equivalent formulations 1Other wordings

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

The toral rank conjecture
Let $X$ be a finite-dimensional CW complex. Define its rational homological rank by
$\mathrm{hrk}(X):=\sum\limits_{i\geqslant 0}\dim H^i(X;\mathbb Q).$
Let $\mathrm{trk}(X)$ denote the maximal rank of a torus acting almost freely on $X$ . The toral rank conjecture. The inequality
$\mathrm{hrk}(X)\geqslant 2^{\mathrm{trk}(X)}$
holds. This conjecture concerns the relationship between almost-free torus actions and the rational cohomology of finite-dimensional CW complexes; its status is not resolved by the supplied source context.
source: Ivan Limonchenko and Grigory Solomadin, “On the homotopy decomposition for the quotient of a moment-angle complex and its applications”, arXiv:2202.13899 (2022).

References

Primary source

Yury Ustinovsky, “Doubling operation for polytopes and torus actions”, arXiv:0909.1050 (2009).

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