Localization genus. (English) Zbl 1370.55006

The extended genus of a connected nilpotent \(CW\)-complex \(X\) of finite type, denoted by \(\overline{G}(X)\), is defined in [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 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 \(\overline{G}_L(X)\) (\(L\)-genus and 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 H. Miller in [Ann. Math. (2) 120, 39–87 (1984; Zbl 0552.55014)]).


55S45 Postnikov systems, \(k\)-invariants
55R15 Classification of fiber spaces or bundles in algebraic topology
55R70 Fibrewise topology
55P20 Eilenberg-Mac Lane spaces
22F50 Groups as automorphisms of other structures
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