## 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)]).

### MSC:

 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

### Citations:

Zbl 0854.55008; Zbl 0552.55014
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