Classification of exterior and proper fibrations.

*(English)*Zbl 1448.55014This technical paper fits in the fields of topological categories and homotopy theory in algebraic topology. Specifically, it concerns some questions in standard proper homotopy theory that are however studied in a much more general setup, that is the category of “exterior” spaces where problems of “exterior” homotopy theory can be solved with the help of methods of standard homotopy theory.

For instance, the category of topological spaces with proper maps can actually be embedded into a larger category which has the advantage of being both complete and co-complete: the category of “exterior spaces”.

Let us now give some definitions. An exterior space is a topological space \(X\) endowed with a nonempty family (subfamily of the topology of \(X\)) of so-called exterior sets which can be thought of as a neighborhood system at infinity. Given two exterior spaces \(X\) and \(Y\), an exterior map is a continuous map from \(X\) to \(Y\) for which the pre-image of any exterior set of \(Y\) is an exterior set of \(X\). A proper fibration is a map in the proper category which is an exterior fibration when considered in the exterior category through the embedding of the category of topological spaces with proper maps into that of exterior spaces.

Finally, there are also the tricky notions of (relative) exterior and based exterior CW-complexes: for what we need now, let us just say that a finite exterior CW-complex is in general a finite dimensional infinite classical CW-complex, while any classical CW-complex is an exterior CW-complex, as well as open manifolds and PL-manifolds (whenever they admit a locally finite countable triangulation).

The aim of the present paper is to classify the set of equivalence classes of exterior fiber sequences over a given exterior path-connected based CW-complex \(X\) with fiber another exterior path-connected based CW-complex \(F\).

This results allows the authors to describe and classify all based proper fibrations between countable, locally finite relative CW-complexes, with fiber \(F\) regarded as an exterior based space.

For instance, the category of topological spaces with proper maps can actually be embedded into a larger category which has the advantage of being both complete and co-complete: the category of “exterior spaces”.

Let us now give some definitions. An exterior space is a topological space \(X\) endowed with a nonempty family (subfamily of the topology of \(X\)) of so-called exterior sets which can be thought of as a neighborhood system at infinity. Given two exterior spaces \(X\) and \(Y\), an exterior map is a continuous map from \(X\) to \(Y\) for which the pre-image of any exterior set of \(Y\) is an exterior set of \(X\). A proper fibration is a map in the proper category which is an exterior fibration when considered in the exterior category through the embedding of the category of topological spaces with proper maps into that of exterior spaces.

Finally, there are also the tricky notions of (relative) exterior and based exterior CW-complexes: for what we need now, let us just say that a finite exterior CW-complex is in general a finite dimensional infinite classical CW-complex, while any classical CW-complex is an exterior CW-complex, as well as open manifolds and PL-manifolds (whenever they admit a locally finite countable triangulation).

The aim of the present paper is to classify the set of equivalence classes of exterior fiber sequences over a given exterior path-connected based CW-complex \(X\) with fiber another exterior path-connected based CW-complex \(F\).

This results allows the authors to describe and classify all based proper fibrations between countable, locally finite relative CW-complexes, with fiber \(F\) regarded as an exterior based space.

Reviewer: Daniele Ettore Otera (Vilnius)

##### MSC:

55P57 | Proper homotopy theory |

55R05 | Fiber spaces in algebraic topology |

55R15 | Classification of fiber spaces or bundles in algebraic topology |

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\textit{J. M. García-Calcines} et al., Proc. Am. Math. Soc. 148, No. 7, 3175--3185 (2020; Zbl 1448.55014)

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##### References:

[1] | Allaud, Guy, On the classification of fiber spaces, Math. Z., 92, 110-125 (1966) · Zbl 0139.16603 |

[2] | Ayala, R.; Dominguez, E.; Quintero, A., Proper homotopy exact sequences for proper fibrations, Rend. Circ. Mat. Palermo (2), 38, 1, 88-96 (1989) · Zbl 0676.55015 |

[3] | Ayala, R.; Quintero, A.; Dominguez, E., A theoretical framework for proper homotopy theory, Math. Proc. Cambridge Philos. Soc., 107, 3, 475-482 (1990) · Zbl 0708.55017 |

[4] | Baues, Hans-Joachim; Quintero, Antonio, Infinite homotopy theory, \(K\)-Monographs in Mathematics 6, viii+296 pp. (2001), Kluwer Academic Publishers, Dordrecht · Zbl 0983.55001 |

[5] | Blomgren, M.; Chach\'{o}lski, W., On the classification of fibrations, Trans. Amer. Math. Soc., 367, 1, 519-557 (2015) · Zbl 1307.55003 |

[6] | Brown, Edgar H., Jr., Abstract homotopy theory, Trans. Amer. Math. Soc., 119, 79-85 (1965) · Zbl 0129.15301 |

[7] | C\'{a}rdenas, Manuel; Lasheras, Francisco F.; Quintero, Antonio, Detecting cohomology classes for the proper LS category. The case of semistable 3-manifolds, Math. Proc. Cambridge Philos. Soc., 152, 2, 223-249 (2012) · Zbl 1245.55002 |

[8] | Chapman, T. A., Proper fibrations with \(n\)-manifold fibers, Indiana Univ. Math. J., 30, 1, 79-102 (1981) · Zbl 0472.55011 |

[9] | Garc\'{\i}a-Calcines, J. M.; Garc\'{\i}a-D\'{\i}az, P. R.; Murillo Mas, A., A Whitehead-Ganea approach for proper Lusternik-Schnirelmann category, Math. Proc. Cambridge Philos. Soc., 142, 3, 439-457 (2007) · Zbl 1143.55007 |

[10] | Garc\'{\i}a-Calcines, Jose M.; Garc\'{\i}a-D\'{\i}az, Pedro R.; Mas, Aniceto Murillo, The Ganea conjecture in proper homotopy via exterior homotopy theory, Math. Proc. Cambridge Philos. Soc., 149, 1, 75-91 (2010) · Zbl 1204.55003 |

[11] | Garc\'{\i}a-Calcines, J. M.; Garc\'{\i}a-D\'{\i}az, P. R.; Murillo, A., Brown representability for exterior cohomology and cohomology with compact supports, J. Lond. Math. Soc. (2), 90, 1, 184-196 (2014) · Zbl 1311.55017 |

[12] | Garc\'{\i}a-Calcines, J. M.; Garc\'{\i}a-Pinillos, M.; Hern\'{a}ndez-Paricio, L. J., A closed simplicial model category for proper homotopy and shape theories, Bull. Austral. Math. Soc., 57, 2, 221-242 (1998) · Zbl 0907.55017 |

[13] | Garcia-Calcines, J. M.; Garcia-Pinillos, M.; Hernandez-Paricio, L. J., Closed simplicial model structures for exterior and proper homotopy theory, Appl. Categ. Structures, 12, 3, 225-243 (2004) · Zbl 1071.55007 |

[14] | Jardine, J. F., Representability theorems for presheaves of spectra, J. Pure Appl. Algebra, 215, 1, 77-88 (2011) · Zbl 1204.55010 |

[15] | Mather, Michael, Pull-backs in homotopy theory, Canadian J. Math., 28, 2, 225-263 (1976) · Zbl 0351.55005 |

[16] | May, J. Peter, Classifying spaces and fibrations, Mem. Amer. Math. Soc., 1, 1, 155, xiii+98 pp. (1975) · Zbl 0321.55033 |

[17] | Sch\"{o}n, Rolf, The Brownian classification of fiber spaces, Arch. Math. (Basel), 39, 4, 359-365 (1982) · Zbl 0512.55014 |

[18] | Stasheff, James, A classification theorem for fibre spaces, Topology, 2, 239-246 (1963) · Zbl 0123.39705 |

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