Quasi-Boolean groups

TL;DR

This work characterizes maximal pro-$2$ quotients of real projective groups in multiple equivalent ways, culminating in a purely cohomological criterion. It shows that such groups are exactly the pro-$2$ groups that are cohomologically quasi-Boolean, and moreover provides an explicit reconstruction $G\cong F(Y)\!*_{2}\mathbb{B}(X)$ from the mod $2$ cohomology ring—where $Y$ indexes a dual part and $X=\mathrm{Spec}(B)$ encodes the Boolean part—via Koszul-style and bar-construction methods. The paper extends Quillen-type theorems to profinite groups, establishes a local-global principle for central embedding problems, and proves that pro-$2$ real projective groups and quasi-Boolean pro-$2$ groups coincide. These results yield concrete presentations and cohomological characterizations with potential applications in Galois theory and fields of low virtual cohomological dimension, connecting topology of profinite spaces with algebraic structure via Stone duality and free pro-$2$ products.

Abstract

We give several equivalent characterisations of the maximal pro-2 quotients of real projective groups. In particular, for pro-2 real projective groups we provide a presentation in terms of generators and relations, and a purely cohomological characterisation. As a consequence we explicitly reconstruct such groups from their mod 2 cohomology rings.

Theorems & Definitions (142)

• Definition 1.1
• Remark 1.2
• Definition 1.3
• Definition 1.4
• Definition 1.5
• Definition 1.6
• Theorem 1.7
• Corollary 1.8
• Theorem 1.9
• Definition 2.1