Dwyer, W. G.; Kan, D. M.; Smith, J. H. Towers of fibrations and homotopical wreath products. (English) Zbl 0683.55018 J. Pure Appl. Algebra 56, No. 1, 9-28 (1989). Given CW-complexes X, \(Y_ 1,\dots,Y_ n\). The problem is to classify, up to homotopy, the towers \[ f: X_ n \overset{f_ n}\longrightarrow X_{n-1} \longrightarrow \cdots \longrightarrow X_ 1 \overset{f_1}\longrightarrow X \] of (Serre) fibrations of CW-complexes such that for all \(i=1,\dots,n\) the fibre of \(f_ i\) has the weak homotopy type of \(Y_ i\). A map \(f\to f'\) of two towers is a commutative diagram \[ \begin{matrix} X_ n & \overset{f_n}\longrightarrow & X_{n-1} & \longrightarrow & \cdots & \longrightarrow & X_1 & \overset{f_1}\longrightarrow & X \\ \\ \downarrow && \downarrow &&&& \downarrow && \downarrow_{\text{id}} \\ \\ X'_ n & \overset{f'_ n}\longrightarrow & X'_{n-1} & \longrightarrow & \cdots & \longrightarrow & X_1' & \overset{f'_1}\longrightarrow & X. \\ \end{matrix} \] If all vertical maps are homotopy equivalences then the map \(f\to f'\) is called a homotopy equivalence of towers. \(BEY_ i\) (resp. BEf) denotes the classifying space of the monoid \(EY_ i\) (resp. Ef) of homotopy self equivalences of the space \(Y_ i\) (resp. of the tower f). The authors construct inductively a homotopical wreath product \(B(EY_ 1,\dots,EY_ n)\) which classifies the towers as above up to homotopy, where the path component of \(B(EY_ 1,\dots,EY_ n)^ X\) determined by f has the homotopy type of BEf. The infinite wreath product \(B(EY_ 1,\dots,EY_ n,\dots)\) is defined by passing to the inverse limit. In the special case of Eilenberg-MacLane spaces \(Y_ i=K(G_ i,i)\), \(i=1,2,\dots\), this limit space classifies up to homotopy the connected CW-complexes Y such that \(\pi_ iY\cong G_ i\), where the component of \(B(EY_ 1,\dots,EY_ n,\dots)^ X\) corresponding to Y has the homotopy type of BEY. Of course the authors actually prove equivalent simplicial results. Reviewer: W.End Cited in 1 ReviewCited in 6 Documents MSC: 55S45 Postnikov systems, \(k\)-invariants 55U10 Simplicial sets and complexes in algebraic topology Keywords:homotopy self equivalences; homotopical wreath product; homotopy type × Cite Format Result Cite Review PDF Full Text: DOI References: [1] Barratt, M. G.; Gugenheim, V. K.A. M.; Moore, J. C., On semisimplicial fibre bundles, Amer. J. Math., 81, 639-657 (1959) · Zbl 0127.39002 [2] Bousfield, A. K.; Kan, D. M., Homotopy Limits, (Localizations and Completions. Localizations and Completions, Lecture Notes in Mathematics, 304 (1972), Springer: Springer Berlin) · Zbl 0259.55004 [3] Dror, E.; Dwyer, W. G.; Kan, D. M., Automorphisms of fibrations, Proc. Amer. Math. Soc., 80, 491-494 (1980) · Zbl 0454.55017 [4] Dwyer, W. G.; Kan, D. M., Homotopy theory and simplicial groupoids, Nederl. Akad. Wetensch. Proc. Ser. A, 379-385 (1984), (87) (=Indag. Math. 46) · Zbl 0559.55023 [5] Dwyer, W. G.; Kan, D. M., Function complexes in homotopical algebra, Topology, 19, 427-440 (1980) · Zbl 0438.55011 [6] Hall, M., The Theory of Groups (1959), MacMillan: MacMillan New York · Zbl 0084.02202 [7] MacLane, S., Categories for the Working Mathematician, (Graduate Texts in Mathematics, 5 (1971), Springer: Springer Berlin) · Zbl 0705.18001 [8] May, J. P., Simplicial Objects in Algebraic Topology (1967), Van Nostrand: Van Nostrand New York · Zbl 0165.26004 [9] Moore, J. C., Semisimplicial complexes and Postnikov systems, Proc. Sympos. Int. Top. Alg., 232-247 (1958), Mexico · Zbl 0089.18001 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.