The homotopy fiber of profinite completion (Q1566930)

For \(Y\) a nilpotent space of finite type with finite fundamental group and Sullivan's profinite completion map \(c:Y\to \widehat{Y}\), the authors show that the basepoint-component of the homotopy fiber of \(c\) is always an \(H\)-space. This observation allows for a reworking of the completion/rationalization approach to phantom map theory. Under this hypothesis and for any connected CW-complex \(X\), they prove \[ Ph(X,Y)\approx \frac{\prod_{n\geq 1}H^n (X; \pi_{n+1} Y\otimes \widehat{Z}/Z)} {[X, \Omega\widehat{Y}]}. \] Using this, they obtain more direct, transparent proofs of various results from the present literature on phantom maps.