  Consider the configuration spaces of manifolds. We give a precise formula for
the integral cohomological dimension (the degree of top non-trivial integral
cohomology group) of unordered configuration spaces of manifolds with
non-trivial co-dimension one cohomology group, and show that the the sequence
of cohomological dimensions is arithmetic. This arithmeticity is not present in
the classical example of Arnold. Moreover, we show that the top integral
cohomology group is infinite. Furthermore, We give a lower bound for the rank
of top integral cohomology group. We also predict that the top integral
cohomology group of configuration spaces of manifolds with non-trivial
co-dimension one cohomology group is eventually finite. To the best of our
knowledge, there is no rigorous bound for the cohomological dimension of
ordered configuration spaces. As an application of main results, we give a
sharp lower bound for the cohomological dimension of ordered configuration
spaces of manifolds. The first step of the proof of main results is to define a
reduced Chevalley Eilenberg complex.
