On Halperin's conjecture relative to torus rank (Q1580686)

Let \(T^r\) be a torus acting almost freely on a finite simply connected CW-complex \(X\). In [`Aspects of topology, Mem. H. Dowker', Lond. Math. Soc. Lect Note Ser. 93, 293-306 (1985; Zbl 0562.57015)], \textit{S. Halperin} conjectured that \(\dim H^*(X; \mathbb{Q})\geq 2^r\). \textit{C. Allday} and \textit{V. Puppe} proved it in some generic cases [Lect. Notes Math. 1217, 1-10 (1986; Zbl 0612.55014)]. In the paper under review, the author proves this conjecture in two particular cases: -- if \(X\) satisfies Poincaré duality and is of codimension \(\leq 6\), -- if \(X\) is the total space of a fibration whose rational cohomology of the fibre and of the loop space of the basis are exterior algebras of finite dimension.