Sheaf theoretic cohomological dimension and finitistic spaces

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.