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A one dimensional manifold is of cohomological dimension $ 2$


Author: Satya Deo
Journal: Proc. Amer. Math. Soc. 52 (1975), 445-446
MSC: Primary 55B30; Secondary 57A65
DOI: https://doi.org/10.1090/S0002-9939-1975-0394632-2
MathSciNet review: 0394632
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Abstract: G. Bredon defines the cohomological Dimension of a topological space $ X$ to be the supremum of all cohomological $ \phi $-dimensions of $ X$, where $ \phi $ varies over the entire families of supports on $ X$. He has proved that if $ X$ is a topological $ n$-manifold then the cohomological Dimension of $ X$ is $ n$ or $ n + 1$. He was not able to decide which one it is, even for a space as simple as the real line. The objective of this paper is to solve his problem for $ n = 1$. In particular, we have shown that the cohomological Dimension of the real line is $ 2$.


References [Enhancements On Off] (What's this?)

  • [1] G. E. Bredon, Sheaf theory, McGraw-Hill, New York, 1967. MR 36 #4552. MR 0221500 (36:4552)
  • [2] R. Godement, Topologie algébrique et théorie des faisceaux, Hermann, Paris, 1958. MR 21 #1583. MR 0102797 (21:1583)
  • [3] A. Grothendieck, Sur quelque points d'algèbre homologique, Tôhoku Math J. (2) 9 (1957), 119-221. MR 21 #1328. MR 0102537 (21:1328)
  • [4] W. Hurewicz and H. Wallmann, Dimension theory, Princeton Math. Ser., vol. 4, Princeton Univ. Press, Princeton, N. J., 1948. MR 3, 312. MR 0006493 (3:312b)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1975-0394632-2
Keywords: Sheaves, sections and serrations of sheaves, extent of a family of supports, paracompactifying family of supports, manifolds and dimension of a topological space
Article copyright: © Copyright 1975 American Mathematical Society

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