A one dimensional manifold is of cohomological dimension $2$

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 $2$.