The integral cohomology groups of configuration spaces of pairs of points in real projective spaces
Jesus Gonzalez, Peter Landweber

TL;DR
This paper computes the integral cohomology groups of configuration spaces of two points in real projective spaces, revealing their structure and applications to symmetric topological complexity.
Contribution
It provides explicit calculations of the integral cohomology of these configuration spaces, linking them to known algebraic structures and completing the topological complexity analysis.
Findings
Explicit cohomology groups for unordered and ordered configurations
Connection to dihedral and elementary abelian 2-groups
Determination of symmetric topological complexity for specific real projective spaces
Abstract
We compute the integral homology and cohomology groups of configuration spaces of two distinct points on a given real projective space. The explicit answer is related to the (known multiplicative structure in the) integral cohomology---with simple and twisted coefficients---of the dihedral group of order 8 (in the case of unordered configurations) and the elementary abelian 2-group of rank 2 (in the case of ordered configurations). As an application, we complete the computation of the symmetric topological complexity of real projective spaces of dimension 2^i + d for d=0,1,2.
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Taxonomy
TopicsHomotopy and Cohomology in Algebraic Topology · Topological and Geometric Data Analysis · Advanced Algebra and Geometry
