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Fibrations and classifying spaces: Overview and the classical examples. (English) Zbl 0991.55010

The paper under review is the fifth in a series of papers by the author [ibid. 34, No. 2, 127-151 (1993; Zbl 0782.55006); 39, No. 2, 83-116 (1998; Zbl 0906.55010), No. 3, 181-203 (1998; Zbl 0917.55009), No. 4, 271-286 (1998; Zbl 0926.55011)]. The objects of study of this series are so-called universal \({\mathcal F}\)-fibrations. Roughly speaking, an \({\mathcal F}\)-fibration should be thought of as a fibration where the fibers are required to be objects in a specified category of spaces \({\mathcal F}\); and a map of \({\mathcal F}\)-fibrations should be thought of as a map that fiberwise is given by a map in \({\mathcal F}\). In the case where the category \({\mathcal F}\) is given by free transitive \(G\)-spaces and \(G\)-maps for a topological group \(G\), the associated \({\mathcal F}\)-fibrations correspond to principal \(G\)-bundles. Other classical notions that can be described in the general set-up of \({\mathcal F}\)-fibrations are principal fibrations, Hurewicz fibrations and sectioned fibrations.
As it is well known, universal principal \(G\)-bundles were first constructed by Milnor. Dold has shown that a principal \(G\)-bundle is universal if and only if the total space of the principal \(G\)-bundle is contractible, which implies that a universal principal \(G\)-bundle also can be obtained by a geometric bar construction. Universal fibrations for Hurewicz or sectioned fibrations with prescribed (pointed) homotopy type have been constructed by various authors. Typically there are two methods that potentially lead to universal \({\mathcal F}\)-fibrations. The first method uses the geometric bar construction; the second method uses the Brown representability theorem. Both methods have advantages and disadvantages. The main goal of this series of papers is to show that under mild assumptions on the category \({\mathcal F}\) both methods can be used to construct a universal \({\mathcal F}\)-fibration. In particular, it follows that there is a universal \({\mathcal F}\)-fibration and moreover it follows that this universal \({\mathcal F}\)-fibration will have all the good properties that can be derived from the geometric bar construction approach, as well as all the good properties that can be derived from the Brown representability approach.
Most of the work that goes into the proof of the main result has been carried out in the preceding papers. This paper essentially provides an overview on what has been achieved, and it concentrates on putting the obtained results into context. In particular, the author dicusses the cases where \({\mathcal F}\)-fibrations correspond to principal \(G\)-bundles and principal fibrations as well as the cases where \({\mathcal F}\)-fibrations correspond to Hurewicz and sectioned fibrations with an arbitrary chosen (pointed) fiber homotopy type. He shows that the main result applies in all of the four cases, which then leads to a very general and satisfactory construction of a corresponding universal fibration.

MSC:

55R65 Generalizations of fiber spaces and bundles in algebraic topology
55R05 Fiber spaces in algebraic topology
55R10 Fiber bundles in algebraic topology
55R15 Classification of fiber spaces or bundles in algebraic topology
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References:

[1] A.G. Allaud , On the classification of fiber spaces , Math. Z. 92 ( 1966 ), 110 - 125 . MR 189035 | Zbl 0139.16603 · Zbl 0139.16603
[2] B1 P. Booth , Local to global properties in the theory of fibrations , Cahiers de Topologie et Géométrie Différentielle Catégoriques , XXXIV - 2 ( 1993 ), 127 - 151 . Numdam | MR 1223656 | Zbl 0782.55006 · Zbl 0782.55006
[3] B2 P. Booth , Fibrations and classifying spaces: an axiomatic approach I , Cahiers de Topologie et Géométrie Différentielle Catégoriques , XXXIX - 2 ( 1998 ), 83 - 116 . Numdam | MR 1631371 | Zbl 0906.55010 · Zbl 0906.55010
[4] B3 P. Booth , Fibrations and classifying spaces: an axiomatic approach II , Cahiers de Topologie et Géométrie Différentielle Catégoriques , XXXIX - 3 ( 1998 ), 181 - 203 . Numdam | MR 1641846 | Zbl 0917.55009 · Zbl 0917.55009
[5] B4 P. Booth , On the geometric bar construction and the Brown representability theorem , Cahiers de Topologie et Géométrie Différentielle Catégoriques , XXXIX - 4 ( 1998 ), 271 - 285 . Numdam | MR 1663342 | Zbl 0926.55011 · Zbl 0926.55011
[6] BHMP P. Booth , P. Heath , C. Morgan and R. Piccinini , H-spaces of self-equivalences of fibrations and bundles , Proc. London Math. Soc. 49 ( 1984 ), 111 - 127 . MR 743373 | Zbl 0525.55005 · Zbl 0525.55005
[7] D1 A. Dold , Partitions of unity in the theory of fibrations , Ann. of Math. 78 ( 1963 ), 223 - 255 . MR 155330 | Zbl 0203.25402 · Zbl 0203.25402
[8] D2 A. Dold , Halbexakte Homotopiefunktoren, Lecture Notes in Math. , vol. 12 , Springer-Verlag , Berlin , 1966 . MR 198464 | Zbl 0136.00801 · Zbl 0136.00801
[9] DL A. Dold and R. Lashof , Principal quasifibrations and fibre homotopy equivalence of bundles , Ill. J. of Math. 3 ( 1959 ), 285 - 305 . MR 101521 | Zbl 0088.15301 · Zbl 0088.15301
[10] FP R. Fritsch and R. Piccinini , Cellular structures in topology , Cambridge University Press , Cambridge , 1990 . MR 1074175 | Zbl 0837.55001 · Zbl 0837.55001
[11] F.M. Fuchs , A modified Dold-Lashof construction that does classify H-principal fibrations , Math. Ann. 192 ( 1971 ), 328 - 340 . MR 292080 | Zbl 0205.27503 · Zbl 0205.27503
[12] M1 J.P. May , Classifying spaces and fibrations , Mem. Amer. Math. Soc. vol. 155 , Providence , 1975 . MR 370579 | Zbl 0321.55033 · Zbl 0321.55033
[13] M2 J.P. May , Fibrewise localization and completion , Trans. Amer. Math. Soc. 258 ( 1980 ), 127 - 146 . MR 554323 | Zbl 0429.55004 · Zbl 0429.55004
[14] Mn C.R.F. Maunder , Algebraic Topology , Van Nostrand Rein-hold , London and New York , 1970 . Zbl 0205.27302 · Zbl 0205.27302
[15] Mi J. Milnor , Construction of universal bundles II , Ann. of Math. 63 ( 1956 ), 430 - 436 . MR 77932 | Zbl 0071.17401 · Zbl 0071.17401
[16] Mo C. Morgan , Characterizations of F-fibrations , Proc. Amer. Math. Soc. 88 ( 1983 ), 169 - 172 . MR 691302 | Zbl 0542.55014 · Zbl 0542.55014
[17] P G.J. Porter , Homomorphisms of principal fibrations: applications to classification, induced fibrations and the extension problem , Ill. J. of Math. 16 ( 1972 ); 41 - 60 . MR 298663 | Zbl 0228.55018 · Zbl 0228.55018
[18] PPP C. Pacati , P. Pavešic and R. Piccinini , On the classification of F-fibrations , Top. and its Applics . 87 ( 1998 ), 213 - 227 . MR 1624324 | Zbl 0928.55020 · Zbl 0928.55020
[19] Sc R. Schön , The Brownian classification of fibre spaces , Arch. Math. 39 ( 1982 ), 359 - 365 . MR 684406 | Zbl 0512.55014 · Zbl 0512.55014
[20] Sib R. Sibson , Existence theorems for H-space inverses , Proc. Camb. Phil. Soc. 65 ( 1969 ), 19 - 21 . MR 233352 | Zbl 0165.26203 · Zbl 0165.26203
[21] Sie J. Siegel , On a space between BH and B\infty , Pacif. J. of Math. 60 , No. 2 , ( 1975 ), 235 - 246 . Article | Zbl 0327.55020 · Zbl 0327.55020
[22] Sp E. Spanier , Algebraic Topology , McGraw-Hill , NewYork , 1966 . MR 210112 | Zbl 0145.43303 · Zbl 0145.43303
[23] Sta J. Stasheff , A classification theorem for fibre spaces , Topology 2 ( 1963 ), 239 - 246 . MR 154286 | Zbl 0123.39705 · Zbl 0123.39705
[24] Str A. STR\emptyset M , The homotopy category is a homotopy category , Arch. der Math. XXIII ( 1972 ), 435 - 441 . · Zbl 0261.18015
[25] Va K. Varadarajan , On fibrations and category , Math. Z. 88 ( 1965 ), 267 - 273 . Article | MR 180969 | Zbl 0163.17905 · Zbl 0163.17905
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