The behavior on fundamental group of a free pro-homotopy equivalence

AbstractWe study the question: given a morphism ƒ{(Xn, xn)}→{(Yn, yn)} in the category pro-(Poi nted. Homotopy) where the domain and range are inverse sequences of well-pointed CW complexes, and given that ƒ induces an isomorphism {Xn}→{Yn} in pro-(Homotopy), what additional hypotheses force ƒ to be an isomorphism in pro-(Pointed Homotopy)? Conjecture. If the dimensions of the Yn's are bounded, then ⧸ is an isomorphism in pro-(Pointed Homotopy). We first prove the special case of this conjecture in which dim Yn⩽d<∞ for all n, and lim {HdYn}≠0, Yn being the universal cover of Yn. Then we deal with the general case: we show that there are certain elements of each π1Yn with the properties: (i) these elements commute if and only if ƒ is an isomorphism in pro-(Pointed Homotopy); (ii) if dim Yn⩽d<∞ for all n, then powers of these elements commute. While (i) and (ii) are not incompatible, this result puts severe restrictions on the nature of any possible counter-example to the conjecture.Our method also gives pro-homotopy analogues of the well-known fact that if a K(π, 1) is N-dimensional, then π is torsion-free and contains no abelian subgroup of rank>N. The latter theorems apply to inverse sequences {Yn} of CW complexes where dim Yn is finite but not necessarily bounded, hence in particular to infinite-dimensional shape-aspherical compacta