The cohomology of free loop spaces of rank 2 flag manifolds

Abstract

By applying Gr\"{o}bner basis theory to spectral sequences algebras, we develop a new computational methodology applicable to any Leray-Serre spectral sequence for which the cohomology of the base space is the quotient of a finitely generated polynomial algebra. We demonstrate the procedure by deducing the cohomology of the free loop space of flag manifolds, presenting a significant extension over previous knowledge of the topology of free loop spaces. A complete flag manifold is the quotient of a Lie group by its maximal torus. The rank of a flag manifold is the dimension of the maximal torus of the Lie group. The rank 2 complete flag manifolds are $SU(3)/T^2$, $Sp(2)/T^2$, $Spin(4)/T^2$, $Spin(5)/T^2$ and $G_2/T^2$. In this paper we calculate the cohomology of the free loop space of the rank 2 complete flag manifolds.

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ORCID for Jelena Grbic: orcid.org/0000-0002-7164-540X

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