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The Ganea conjecture in proper homotopy via exterior homotopy theory

  • JOSE M. GARCÍA–CALCINES (a1), PEDRO R. GARCÍA–DÍAZ (a1) and ANICETO MURILLO MAS (a2)
Abstract

In this article we provide sufficient conditions on a space X to verify Ganea conjecture with respect to exterior and proper Lusternik–Schnirelmann category. For this aim we previously develop an exterior version of the Whitehead, cellular approximation, CW-approximation and Blakers–Massey theorems within a homotopy theory of exterior CW-complexes and study their corresponding analogues and consequences in the proper setting.

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[1]Ayala, R., Domínguez, E. and Quintero, A.A theoretical framework for proper homotopy theory. Math. Proc. Camb. Phil. Soc. 107 (1990), 475482.
[2]Ayala, R., Domínguez, E., Márquez, A. and Quintero, A.Lusternik–Schnirelmann invariants in proper homotopy theory. Pacific J. Math. 153 (1992), 201215.
[3]Ayala, R. and Quintero, A.On the Ganea strong category in proper homotopy. Proc. Edinburgh Math. Soc. 41 (1998), 247263.
[4]Baues, H. J., Foundations of proper homotopy theory. Max-Planck Institut fr Mathematik, preprint (1992).
[5]Baues, H. J. and Quintero, A.Infinite homotopy theory. K-Monographs in Mathematics 6, (Kluwer Academic Publishers, 2001).
[6]Baues, H. J. and Quintero, A.On the locally finite chain algebra of a proper homotopy type. Bull. Belg. Math. Soc. 3:(2) (1996), 161175.
[7]Brown, E. M.On the proper homotopy type of simplicial complexes. Lect. Notes in Math. 375 (1975).
[8]Cárdenas, M., Lasheras, F. F. and Quintero, A.Minimal covers of open manifolds with half-spaces and the proper L-S category of product spaces. Bull. Belgian Math. Soc. 9 (2002), 419431.
[9]Cárdenas, M., Lasheras, F. F., Muro, F. and Quintero, A.Proper L-S category, fundamental pro-groups and 2-dimensional proper co-H-spaces. Topology Appl. 153:(4) (2005), 580604.
[10]Cárdenas, M., Muro, F. and Quintero, A.The proper L-S category of Whitehead manifolds. Topology Appl. 153:4, (2005), 557579.
[11]Cornea, O., Lupton, G., Oprea, J. and Tanré, D.Lusternik–Schnirelmann category. Math. Surveys and Monogr. 103, (Amer. Math. Soc., 2003).
[12]Doeraene, J. P.L. S.-category in a model category. J. Pure Appl. Alg. 84 (1993), 215261.
[13]Edwards, D. and Hastings, H.Čech and Steenrod homotopy theories with applications to Geometric Topology. Lect. Notes Math. 542 (Springer, 1976).
[14]Extremiana, J. I., Hernández, L. J. and Rivas, M. T.Postnikov factorizations at infinity. Topology Appl. 153 (2005), 370393.
[15]Extremiana, J. I. and Hernández, L. J. y Rivas, M. T.Proper CW-complexes: a category for the study of proper homotopy. Collect. Math., 39:(2), (1988), 149179.
[16]Farrell, F. T., Taylor, L. R. y Wagoner, J. B.The Whitehead theorem in the proper category. Compositio Math., 27 (1973),1–23.
[17]García–Calcines, J. M., García–Díaz, P. R. and Murillo–Mas, A.A Whitehead-Ganea approach for proper Lusternik–Schnirelmann category. Math. Proc. Camb. Phil. Soc. 142:(3) (2007), 439457.
[18]García–Calcines, J. M., García–Pinillos, M. and Hernández–Paricio, L. J.A closed model category for proper homotopy and shape theories. Bull. Austral. Math. Soc. 57:(2) (1998), 221242.
[19]García–Calcines, J. M., García–Pinillos, M. and Hernández–Paricio, L. J.Closed simplicial model structures for exterior and proper homotopy theory. Appl. Categ. Structures 12 (3) (2004), 225243.
[20]García–Díaz, P. R. Caracterizaciones de Whitehead y Ganea para la categoría de Lusternik-Schnirelmann propia. Tesis (Spanish), (2007).
[21]Gray, B.Homotopy Theory: An introduction to algebraic topology. Pure Appl. Math. 64 (1975).
[22]Hernández–Paricio, L. J.Application of simplicial M-sets to proper homotopy and strong shape theories. Trans. Amer. Math. Soc. 347:(2) (1995), 363409.
[23]Hernández–Paricio, L. J.Functorial and algebraic properties of Brown's functor. Theory Appl. Categ. 1:(2) (1995), 1053.
[24]Iwase, N.Ganea conjecture on Lusternik-Schnirelmann category. Bull. London Math. Soc. 30:(6) (1998), 623634.
[25]Strøm, A.The homotopy category is a homotopy category. Arch. Math. 23 (1972), 435441.
[26]Vandembroucq, L.Suspension of Ganea fibrations and a Hopf invariant. Topology Appl. 105 (2000), 187200.
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Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
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