The Integral Homology Ring of the Based Loop Space on some Generalised Symmetric Spaces

Abstract

In this paper we compute the integral Pontrjagin homology ring of the based loop spaceon some generalised symmetric spaces G/S, where G is a simple compact Lie group Ghaving the torus S as a stationary subgroup. More precisely, we consider generalisedsymmetric spaces which have zero Euler characteristic, that is, for which the torus S is notmaximal in G. Thus this paper can be consider as a natural sequel to paper [5], where theauthors computed the integral Pontrjagin homology of the based loop space of completeflag manifolds of compact simple Lie groups.Generalised symmetric spaces G/H are defined by the condition that their stationarysubgroup H is the fixed point subgroup of a finite order automorphism of th e group G.These spaces consequently admit finite order symmetries mak ing them to have rich ge-ometry and thus attract constant attention of many geometry focused research startingwith [14],[15] until recent once, see for example [1],[2],[7]. Generalised symmetric spacesplayimportantrolein the theoryof homogeneousspaces and, thus, in manyareas of mathe-matics and physics such as representation theory, combinatorics, string topology. In partic-ular complex homogeneous spaces, which are important examples in complex cobordismsand theory of characteristic classes, can be described by complex generalised symmetricspaces.The methods used in this paper are analogous to that of [5]. By [12] all generalisedsymmetric spaces are Cartan pair homogeneous spaces and, thus, formal in the sense ofSullivan. Therefore,startingfromtheirrationalcohomologyalgebrasandusingthe methodof rational homotopy theory together with the Milnor and Moore theorem which expressesthe rational Pontrjagin homology algebra in terms of the universal enveloping algebra ofthe rational homotopy Lie algebra, we compute the rational homology algebras for thebased loop on generalised symmetric spaces G/S. We prove that the integral loop ho-mology groups of these spaces have no torsion which together with the rational computa-tions enables us to determine their integral Pontrjagin ring structure as well. In particular,we calculate the integral Pontrjagin homology ring of the based loop spaces on the fol-lowing generalised symmetric spaces: SU(2n + 1)/T