Complementation of relations in pro-p-groups of cohomology dimension two

Abstract

It is proved that if, from the minimal corepresentation of a pro-p-group ~ of cohomology dimension two with a free commutant, a part of the relations is discarded, then one obtains a minimal corepresentation of a pro-p-group, the cohomology dimension of which is equal to two and the commutant of which is free. For such pro-pgroups, F(~) < ~(~) , where ~) is the minimal number of generators of the finitely generated pro-p-group ~ and ~ [~) is the minimal number of relations.Conversely, if ~ is a finitely generated pro-p-group of cohomology dimension two with a free conmmtant, then, for it, any minimal system of relations can be complemented to give a system of relations such that the pro-p-group ~/ thus obtained has c0homology dimension two, its commutant is free, and, moreover, ~(~I)= ~(~t)-1 =~G) -~

Rational and $p$-adic analogs of J.H.C. Whitehead's conjecture

“…In Section 5.4 we adapt V.M.Tsvetkov's ideas [53] about pro-p-groups of cohomological dimension 2, obtaining a criterium for asphericity for subpresentations, we also replace his dubious argument in the proof of equivalence (a), (b) and (d), (e) on the spectral sequence argument prepared in Section 3.3. Section 5.5 contains the proof of Theorem 1.…”