2023/05/11 by Muhammad Yameen, Yameen, Muhammad
Computer Science · Mathematics · #Advanced Topics in Algebra #Algebraic Geometry (math.AG) #Algebraic Topology (math.AT) #FOS: Mathematics #Geometric Topology (math.GT) #Homotopy and Cohomology in Algebraic Topology #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.2305.06604
openalex publication_date 2023/05/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.