The cohomology of free loop spaces of rank $2$ flag manifolds

By applying Gröbner 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 $SU(3)/T^2$, $Sp(2)/T^2$, $\mathit{Spin}(4)/T^2$, $\mathit{Spin}(5)/T^2$ and $G_2/T^2$. In this paper we calculate the cohomology of the free loop space of the rank $2$ complete flag manifolds.

Keywords

cohomology, free loop space, flag manifold, spectral sequence, homotopy theory, Gröbner basis

2010 Mathematics Subject Classification

13P10, 14M15, 55P35, 55T10

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This research was supported in part by The Leverhulme Trust Research Project Grant RPG-2012-560.

Received 6 March 2022

Received revised 19 September 2022

Accepted 4 November 2022

Published 22 November 2023