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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

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 $2$.


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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