×

Exact sequences and closed model categories. (English) Zbl 1204.55016

The authors construct Eckman-Hilton and Puppe sequences in model categories without zero object using the category over the initial object and the category under the final object. As an application, the authors obtain that the cohomology of a group \(G\) with coefficients in a \(G\)-module can be interpreted as certain homotopy groups of twisted Eilenberg-Mac Lane spaces in the category of spaces under and over the Eilenberg-Mac Lane space \(K(G,1)\). Finally, an application in the category of exterior spaces under \(\mathbb{R}_+\) is given, connecting Brown-Grossmann exterior homotopy groups with Steenrod exterior homotopy groups by means of an exact sequence.

MSC:

55U35 Abstract and axiomatic homotopy theory in algebraic topology
55U40 Topological categories, foundations of homotopy theory
55N25 Homology with local coefficients, equivariant cohomology
55Q70 Homotopy groups of special types
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Alvarez, V., Armario, J.A., Real, P.: On the computability of the p-local homology of twisted cartesian products of Eilenberg-Mac Lane spaces. In: First Meeting on Geometry and Topology, Braga, 11–13 September 1997 · Zbl 0941.55007
[2] Baues, H.J.: Algebraic Homotopy. Cambridge University Press, Cambridge (1988) · Zbl 0673.55007
[3] Baues, H.J., Quintero, A.: Infinite homotopy theory. K-monogr. Math. 6 (2001) · Zbl 0983.55001
[4] Brown, K.S.: Cohomology of Groups. GTM, vol. 87. Springer, New York (1982)
[5] Brown, E.M.: On the proper homotopy type of simplicial complexes. Topology Conference, Lect. Notes in Math., no. 375, pp. 41–46. Springer, New York (1975)
[6] Brown, E.M., Tucker, T.W., On proper homotopy theory for non compact 3-manifolds. Trans. Amer. Math. Soc. 188(7), 105–126 (1977) · Zbl 0289.57002
[7] Casacuberta, C., Hernández L.J., Rodríguez J.L.: Models for torsion homotopy types. Israel J. Math. 107, 301–318 (1998) · Zbl 0916.55012 · doi:10.1007/BF02764014
[8] Cordier, J.M., Porter, T.: Shape Theory, Categorical Methods of Approximation. Ellis Horwood Series in Math and its Applications. Ellis Horwood, Chichester (1989) · Zbl 0663.18001
[9] Dwyer, W.G., Spalinski, J.: Homotopy theories and model categories. In: Handbook of Algebraic Topology, pp. 73–126. Elsevier, Amsterdam (1995) · Zbl 0869.55018
[10] Dwyer, W.G., Hirschhorn, P.S., Kan, D.M., Smith, J.: Homotopy limit functors on model categories and homotopical categories. In: Mathematical Surveys Monographs, vol. 113, 181 pp. (2005) · Zbl 1072.18012
[11] Edwards, D., Hastings, H.: Čech and Steenrod homotopy theories with applications to geometric topology. In: Lectures Notes in Math, vol. 542. Springer, New York (1976) · Zbl 0334.55001
[12] Extremiana, J.I., Hernández, L.J., Rivas, M.T.: An isomorphism theorem of the Hurewicz type in the proper homotopy category. Fund. Math. 132, 195–214 (1989) · Zbl 0684.55014
[13] Extremiana, J.I., Hernández L.J., Rivas, M.T: A closed model category for (n1)-connected spaces. Proc. Amer. Math. Soc. 124, 3545–3553 (1996) · Zbl 0866.55008 · doi:10.1090/S0002-9939-96-03606-4
[14] Freudenthal, H.: Über die Enden topologisher Räume und gruppen, Math. Z. 53, 692–713 (1931) · Zbl 0002.05603 · doi:10.1007/BF01174375
[15] Freedman, M.H.: The topology of four-dimensional manifolds. J. Differential Geom. 17, 357–453 (1982) · Zbl 0528.57011
[16] Gabriel, P., Zisman, M.: Calculus of Fractions and Homotopy Theory. Springer, Berlin (1966) · Zbl 0186.56802
[17] García, J.M., García, M., Hernández, L.J.: A closed simplicial model category for homotopy and shape theories. Bull. Austral. Math. Soc. 57, 221–242 (1998) · Zbl 0907.55017 · doi:10.1017/S0004972700031610
[18] García, J.M., García, M., Hernández L.J.: Closed simplicial model structures for exterior and proper homotopy theory. Appl. Categ. Structures 12(3), 225–243 (2004) · Zbl 1071.55007 · doi:10.1023/B:APCS.0000031087.83413.a4
[19] García-Calcines, J.M., García-Díaz, 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 · doi:10.1017/S0305004106009960
[20] Goerss, P.G., Jardine, J.F.: Simplicial Homotopy Theory. Progr. Math., vol. 174. Birkhäuser Verlag, Basel (1999) · Zbl 0949.55001
[21] Grossman, J.W.: Homotopy groups of pro-spaces. Ill. J. Math. 20, 622–625 (1976) · Zbl 0329.55011
[22] Hernández, L.J.: Application of simplicial M-sets to proper homotopy and strong shape theories. Trans. Amer. Math. Soc. 347(2), 363–409 (1995) · Zbl 0855.55007 · doi:10.2307/2154894
[23] Hernández, L.J.: Closed model categories for uniquely S-divisible spaces. J. Pure Appl. Algebra 182, 223–237 (2003) · Zbl 1024.55007 · doi:10.1016/S0022-4049(03)00016-1
[24] Hirschhorn, P.S.: Model categories and their localizations. In: Mathematical Surveys Monographs, vol. 99, 457 pp. (2003) · Zbl 1017.55001
[25] Hovey, M.: Model categories. In: Mathematical Surveys and Monographs, Providence, vol. 63, pp. x + 209 pages. AMS, Providence (1999) · Zbl 0909.55001
[26] Mardešić, S., Segal, J.: Shape Theory. North-Holland, Amsterdam (1982)
[27] Møller, J.M.: Spaces of sections of Eilenberg-Mac Lane fibrations. Pacific J. Math. 130, 171–186 (1987) · Zbl 0599.55010
[28] Porter, T.: Proper homotopy theory. In: James, I.M. (ed.) Handbook of Algebraic Topology (ch. 3), pp. 127–167. North Holland, Amsterdam (1995) · Zbl 1004.55004
[29] Porter, T.: Čech and Steenrod homotopy and the Quigley exact couple in strong shape and proper homotopy theory. J. Pure Appl. Algebra 24, 303–312 (1983) · Zbl 0485.55009 · doi:10.1016/0022-4049(82)90049-4
[30] Siebenmann, L.C.: The obstruction to finding a boundary for an open manifold of dimension greater than five. Ph.D. thesis (1965)
[31] Siebenmann, L.C.: Infinite simple homotopy types. Indag. Math. 32, 479–495 (1970) · Zbl 0203.56002
[32] Quillen, D.: Homotopical algebra. In: Lectures Notes in Math, vol. 43. Springer, New York (1967) · Zbl 0168.20903
[33] Quillen, D.: Rational homotopy theory. Ann. of Math. 90, 205–295 (1969) · Zbl 0191.53702 · doi:10.2307/1970725
[34] Quigley, J.B.: An exact sequence from the n-th to the (n-1)-st fundamental group. Fund. Math. 77, 195–210 (1973) · Zbl 0247.55010
[35] Tsuchida, K.: Generalized Puppe sequence and Spanier-Whitehead duality. Tôcutu Math. J. 23, 37–48 (1971) · Zbl 0212.28205 · doi:10.2748/tmj/1178242685
[36] Whitehead, G.W.: Elements of Homotopy Theory. Springer, New York (1995) · Zbl 0829.05048
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.