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Sheaf theoretic cohomological dimension and finitistic spaces

Author: Satya Deo
Journal: Proc. Amer. Math. Soc. 86 (1982), 545-550
MSC: Primary 55N30; Secondary 54F45, 55M10
MathSciNet review: 671233
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Abstract: For a topological $ n$-manifold $ X$, we proved earlier [7] that $ {\text{Di}}{{\text{m}}_{\mathbf{Z}}}(X) = n + 1$, if $ n > 0$; and, for a zero-dimensional manifold (discrete space) we observed that $ {\text{Di}}{{\text{m}}_{\mathbf{Z}}}(X) = 0$. The question was later raised as to what are those paracompact spaces, besides discrete one, for which $ {\text{Di}}{{\text{m}}_{\mathbf{Z}}}(X) = 0$. In this paper we prove that there is none, i.e., if $ X$ is not discrete then $ {\text{Di}}{{\text{m}}_{\mathbf{Z}}}(X) > 0$. Another question which cropped up only recently in the cohomological theory of topological transformation groups is whether or not there exists a finitistic space which is not of finite (sheaf theoretic) integral cohomological dimension. We show that this question is related to a famous unsolved problem of cohomological dimension theory.

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Keywords: Sheaves, family of supports, sheaf theoretic cohomological dimension, real-compactness, finitistic spaces
Article copyright: © Copyright 1982 American Mathematical Society