Connective coverings, phantom maps and genus sets C. McGibbonJoseph Roitberg connective coveringphantom mapsMislin genuscompletion genus Let $X\langle n\rangle$ denote the $n$-connective covering of a space $X$. For a number of familiar finite type $CW$-complexes $X$, $Y$, we study: (i) the set $\text{Ph}(X\langle n\rangle, Y)$, consisting of homotopy classes of phantom maps from $X\langle n\rangle$ to $Y$, together with the group structure on this set when $Y$ is a rational $H$-space; (ii) the genus set $\mathcal{G}(X\langle n\rangle)$, consisting of all homotopy types of finite type $CW$-complexes $p$-equivalent to $X\langle n\rangle$ for all primes $p$. Indiana University Mathematics Journal 1998 text pdf 10.1512/iumj.1998.47.1586 10.1512/iumj.1998.47.1586 en Indiana Univ. Math. J. 47 (1998) 1433 - 1458 state-of-the-art mathematics http://iumj.org/access/