Localization genus (Q517526)

The \textit{extended genus} of a connected nilpotent \(CW\)-complex \(X\) of finite type, denoted by \(\overline{G}(X)\), is defined in [\textit{C. A. McGibbon}, Progr.\ Math.\ 136, 285--306 (1994; Zbl 0854.55008)] as the set of homotopy types \([Y]\) of nilpotent CW-spaces \(Y\) such that \(X_{(p)}\simeq Y_{(p)}\) for all primes \(p\) (\(X_{(p)}\) denotes the localization of \(X\) at prime \(p\)). This extends the \textit{Mislin genus} \(G(X)\) which is the subset of \(\overline{G}(X)\) consisting of \([Y]\) with \(Y\) of finite type. In the paper under review the authors generalize these notions and, for a given localization functor \(L\) and a simply connected \(CW\)-complex \(X\) of finite type, they introduce \(G_L(X)\) and \(\overline{G}_L(X)\) (\textit{\(L\)-genus} and \textit{extended \(L\)-genus} of \(X\)), which coincide with \(G(X)\) and \(\overline{G}(X)\) when \(L\) is the rationalization functor. In [loc. cit.] it is shown that \(\overline{G}(S^n)\) is uncountable if \(n\) is odd. A significant result of the present paper is the assertion that there is a bijection between \(\overline{G}(S^n)\) and the set of isomorphism classes of torsion free abelian groups of rank one. Also, the authors present some computations concerning the extended Postnikov genus set \(\overline{G}_{[n]}(S^n)\) (for \(n\) odd), Neisendorfer localization and the Sullivan conjecture (resolved affirmatively by \textit{H. Miller} in [Ann.\ Math.\ (2) 120, 39--87 (1984; Zbl 0552.55014)]).