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The Urysohn-Menger sum formula: an extension of the Dydak-Walsh theorem to dimension one

Part of: Homotopy theory Classical Topics in Algebraic Topology Special properties

Published online by Cambridge University Press:  09 April 2009

Aleksander N. Dranišnikov  and
Dušan Repovš
Aleksander N. Dranišnikov
Affiliation:
Department of Mathematics, Cornell University, White Hall, Ithaca, NY 14853-7901, USA, e-mail: dranish@math.cornell.edu
Dušan Repovš
Affiliation:
Institute for Mathematics Physics and Mechanics, University of Ljubljana, P. O. Box 64 Ljubljana 61111, Slovenia, e-mail: dusan.repovs@uni-lj.si
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Abstract

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Let X be a finite-dimensional separable metric space, presented as a disjoint union of subsets, X = A∪B. We prove the following theorem: For every prime p, c-dimZpX≦c-dimZpA + c–dimZpB + 1. This improves upon some of the earlier work by Dydak and Walsh.

Keywords

Urysohn-Menger sum formulacohomological dimensionfinite groupsBock-stein ringsMoore spacesEilenberg-MacLane complexesBlakers-Massey theorem.

MSC classification

Secondary: 55M10: Dimension theory 55P20: Eilenberg-Mac Lane spaces 54F45: Dimension theory 55P99: None of the above, but in this section

Information

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 59 , Issue 2 , October 1995 , pp. 273 - 282
Copyright
Copyright © Australian Mathematical Society 1995

References

[1]Dranišnikov, A. N., ‘On a problem of P. S. Aleksandrov’, Mat. Sbornik (177) 135 (177) (1988), 551–557 (in Russian). English translation Math. USSR Sbornik 63 (1988), 539–545.Google Scholar
[2]Dranišnikov, A. N., ‘Homological dimension theory’, Uspekhi. Mat. Nauk 43:4 (1988), 11–55 (in Russian). English translation Russian Math. Surv. 43:4 (1988), 11–63.Google Scholar
[3]Dranišnikov, A. N., ‘On intersections of compacta in Euclidean space, I’, Proc. Amer. Math. Soc. 112 (1991), 267–275.CrossRefGoogle Scholar
[4]Dranišnikov, A. N., ‘Extension of mappings into CW complexes’, Mat. Sbornik 182 (1991), 1300–1310 (in Russian). English translation Math. USSR Sbornik 74 (1993) 47–56.Google Scholar
[5]Dranišnikov, A. N. and Repovš, D., ‘The Urysohn-Menger sum formula for cohomological dimension: An improvement of the Dydak-Walsh theorem’, Abstracts Amer. Math. Soc. 14 (1993), 219, No. 93T–55–43.Google Scholar
[6]Dranišnikov, A. N., Repovš, D. and Ščepin, E. V., ‘On the failure of the Urysohn-Menger sum formula for cohomological dimension’, Proc. Amer. Math. Soc. 120 (1994), 1267–1270.CrossRefGoogle Scholar
[7]Dydak, J., ‘Cohomological dimension and metrizable spaces’, Trans. Amer. Math. Soc. 337 (1993), 219–234.CrossRefGoogle Scholar
[8]Dydak, J., ‘Cohomological dimension and metrizable spaces, II.’, Trans. Amer. Math. Soc., to appear.Google Scholar
[9]Dydak, J., ‘Union theorem for cohomological dimension: A simple counterexample’, Proc. Amer. Math. Soc. 121 (1994), 295–297.CrossRefGoogle Scholar
[10]Dydak, J. and Walsh, J. J., ‘Aspects of cohomological dimension for principal ideal domains’, preprint, Univ. of Tennessee, Knoxville 1992.Google Scholar
[11]Engelking, R., Dimension theory (North-Holland, Amsterdam, 1975).Google Scholar
[12]Gray, B., Homotopy theory: an introduction to algebraic topology (Academic Press, New York, 1975).Google Scholar
[13]Kuz'minov, V. I., ‘Homological dimension theory’, Uspekhi Mat. Nauk 23:5 (1968), 3–49 (in Russian). English translation Russian Math. Surveys 23:5 (1968), 1–45.Google Scholar
[14]Mitchell, W. J. R. and Repovš, D., ‘The topology of cell-like mappings’, in: Proc. Conf. Diff. Geom. Topology Cala Gonone 1988, Suppl. Rend. Sem. Fac. Sci. Univ. Cagliari 58 (1988), 265–300.Google Scholar
[15]Rubin, L. R., ‘Characterizing cohomological dimensions: The cohomological dimension of A∪B’, Topology Appl. 40 (1992), 233–263.CrossRefGoogle Scholar
[16]Walsh, J. J., ‘Dimension, cohomological dimension and cell-like mappings’, in: Shape theory and geometric topology, Dubrovnik 1981 (eds. Mardešić, S. and Segal, J.), Lecture Notes in Math. 870 (Springer, Berlin, 1981) pp. 105–118.Google Scholar