Small covers and the Halperin-Carlsson conjecture (Q433559)

The paper under review deals with the Halperin-Carlsson conjecture, saying that if \(G=({\mathbb Z}_p)^m\) or \((S^1)^m\) can act freely on a finite CW-complex \(X\), then, respectively, NEWLINE\[NEWLINE\sum_{i=0}^\infty \dim_{{\mathbb Z}_p}H^i(X;{\mathbb Z}_p)\geq 2^m \text{ or } \sum_{i=0}^\infty \dim_{{\mathbb Q}}H^i(X;{\mathbb Q})\geq 2^m.NEWLINE\]NEWLINE The contribution of the paper under review proves the Halperin-Carlsson conjecture for any free \(({\mathbb Z}_2)^m\)-action on a compact smooth manifold whose orbit space is a small cover. The method of proof is based upon the reconstruction of a principal \(({\mathbb Z}_2)^m\)-bundle over \(Q^n\) via a standard glue-back construction from \(Q^n\), developed by the author of the paper in [\textit{L. Yu}, Osaka J. Math. 49, No. 1, 167--193 (2012; Zbl 1245.57031)].