On the computation of homology of type A real flag manifolds

TL;DR

This paper provides a closed formula for computing the cellular homology of type A real flag manifolds, including explicit calculations for higher homology groups, advancing the understanding of their topological structure.

Contribution

It introduces a new, explicit formula for cellular homology coefficients of real flag manifolds of type A, with detailed computations for higher homology groups.

Findings

Derived a closed, computable formula for homology coefficients

Computed third and fourth homology groups explicitly

Identified generators for the free part up to sixth homology

Abstract

In this paper, we present a closed, computable formula for the cellular homology coefficients of real flag manifolds associated with split real forms of type A. We demonstrate the process using movements within the code diagram for permutations. Additionally, we compute the third and fourth homology groups and provide generators for the free part up to the sixth homology group.