The integral homology ring of the based loop space on some generalised symmetric spaces

TL;DR

This paper computes the integral Pontrjagin homology ring of the based loop space on certain generalized symmetric spaces with a toral stationary subgroup, extending the method to other types.

Contribution

It introduces a method to calculate the homology ring of loop spaces on generalized symmetric spaces, applicable to various types.

Findings

Computed the homology ring for specific generalized symmetric spaces.

Demonstrated the method's applicability to other types in the appendix.

Provided explicit algebraic structures of the homology rings.

Abstract

In this paper we calculate the integral Pontrjagin homology ring of the based loop space on some generalised symmetric spaces with a toral stationary subgroup. In the Appendix we show that the method can be applied to other type generalised symmetric spaces as well.