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

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 1

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).

Sources & referencesView supporting material

Primary source

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

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The toral rank conjecture

Equivalent formulations 1

The toral rank conjecture

Primary source

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