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Local homology and cohomology

Published online by Cambridge University Press:  24 October 2008

W. J. R. Mitchell
W. J. R. Mitchell
Affiliation:
Magdalene College, Cambridge
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Extract

The idea of associating local homology and cohomology groups to each point of a space is now over fifty years old, and it has proved useful in the theory of transformation groups (as in the local Smith theorems), in dimension theory and elsewhere. Nevertheless there is no uniformity as to definition or notation, and a good deal of the theory is written in a notation now superseded and often impenetrable to the modern reader. In this paper all the important definitions are collected in a single framework, and the relations between the various groups are explored. Particular attention is paid to the case of a cohomologically locally connected space, for which relatively complete results are obtained.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 89 , Issue 2 , March 1981 , pp. 309 - 324
Copyright
Copyright © Cambridge Philosophical Society 1981

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References

(1)Alexandroff, P.Dimensionstheorie. Math. Annalen 106 (1932), 161–238.Google Scholar
(2)Alexandroff, P.On the Local Properties of Closed Sets. Annals of Math. 36 (1935), 1–35.Google Scholar
(3)Artin, M. and Mazur, B.Etale Homotopy. Lecture Notes in Mathematics no. 100 (Springer 1969).Google Scholar
(4)Beniaminov, E. M. and Skljarenko, E. G.Local Cohomology Groups. dokl. Akad. Nauk. SSSR 176 (1967), 987–990. Sov. Math. Doklady 8 (1967), 1221–1225.Google Scholar
(5)Bing, R. H. and Borsuk, K.Locally Homogeneous Spaces. Annals of Math. 81 (1965), 100–111.Google Scholar
(6)Borel, A. et al. Seminar on Transformation Groups. Annals of Math. Study no. 46 (Princeton 1960).Google Scholar
(7)Borel, A.The Poincaré Duality in Generalised Manifolds. Michigan Math. J. 4 (1957), 227–239.CrossRefGoogle Scholar
(8)Borel, A. and Moore, J. C.Homology Theory for Locally Compact Spaces. Michigan Math. J. 7 (1960), 137–159.Google Scholar
(9)Bourbaki, N.General Topology, Part 1. (English Translation.) Addison-Wesley 1966.Google Scholar
(10)Bredon, G. E.Sheaf Theory. McGraw-Hill 1967.Google Scholar
(11)Bredon, G. E.Generalised Manifolds Revisited. Proceedings of the Georgia Conference 1969. Ed. Cantrell, J. C. and Edwards, C. H. (Markham 1970).Google Scholar
(12)Edwards, D. A. and Geoghegan, R.Compacta Weak Shape Equivalent to ANRs. Fund. Math. 90 (1975), 115–124.Google Scholar
(13)Fuchs, L.Infinite Abelian Groups, volume 1. Academic Press 1970.Google Scholar
(14)Harlap, A. E.Local Homology and Cohomology, Homology Dimension and Generalised Manifolds. Mat. Sbornik 96 (1975), 347–373. Math. USSR Sbornik 25 (1975), 323–349.Google Scholar
(15)Hill, P.Limits of Sequences of Finitely Generated Abelian Groups. Proc. Amer. Math. Soc. 12 (1961), 946–950.Google Scholar
(16)Jensen, C. U.Les Foncteurs Dérivés de Km et leurs Applications en Théorie des Modules. Lecture Notes in Maths, no. 254 (Springer 1972).Google Scholar
(17)Laudal, O. A.Cohomologie Locale. Applications. Math. Scand. 12 (1963), 147–162.Google Scholar
(18)Milnor, J. W.On Axiomatic Homology Theory. Pacific J. Math. 12 (1962), 337–341.Google Scholar
(19)Mirjuk, V. J.Homology and Cohomology of Sets and their Neighbourhoods. Mat. Sbornik 92 (134) (1973), 306–318. Math. USSR Sbornik 21 (1973), 303–316.Google Scholar
(20)Mitchell, W. J. R.Homology Manifolds, Inverse Systems and Cohomological Local Connectedness. J. London Math. Soc. (2), 19 (1979), 348–358.Google Scholar
(21)Mitchell, W. J. R. Constancy Modulo a Serre Class, with Applications. (In preparation.)Google Scholar
(22)Mitchell, W. J. R.Dual Modules. Math. Proc. Cambridge Philos. Soc. 84 (1978), 21–24.Google Scholar
(23)Raymond, F.Local Cohomology Groups with Closed Support. Math. Z. 76 (1961), 31–41.Google Scholar
(24)Skljarenko, E. G.Homology Theory and the Exactness Axiom. Uspekhi Mat. Nauk. 24 (1969), no. 5 (149), 87–140. Russian Maths. Surveys 24 (1969), no. 5, 91–142.Google Scholar
(25)Verdier, J. L.Equivalence essentielle des systèmes projectifs. C. R. Acad. Sci. Paris 261 (1965), 4950–4953.Google Scholar
(26)Verdier, J. L.Faisceaux constructibles sur un espace localement compact. G. R. Acad. Sci. Paris Sér. A–B 262 (1966), A 12–15.Google Scholar
(27)Wilder, R. L.Topology of Manifolds. Amer. Math. Soc. Colloq. Pub. 32 (1949).Google Scholar
(28)Geoghegan, R.A Note on the Vanishing of Lim1. J. Pure and Applied Algebra 17 (1980), 113–116.Google Scholar