Cohomology of loop spaces of Hermitian symmetric spaces of classical types (Q1935836)

Let \(M\) be an irreducible compact Hermitian space of classical type. Then it is well-know that the space \(M\) is one of following spaces \[ \begin{aligned} & G_{m,n}(\mathbb C)=U(m+n)/(U(m)\times U(n))\quad (m,n\geq 1)\tag{AIII} \\ & Q_n=SO(n+2)/(SO(2)\times SO(n))\quad (n\geq 3)\tag{BDI} \\ & Sp(n)/U(n)\quad (n\geq 3)\tag{CI} \\ & SO(2n)/SO(n) \quad (n\geq 4)\tag{DIII} \end{aligned} \] In this paper, the author studies the mod \(p\) cohomology of the loop space \(\Omega M\) and of the free loop space \(LM\), and he proves that the integral cohomology of them is torsion free, or has only \(2\)-torsion of order \(2\) by computing them explicitly. He also considers the Serre spectral sequence of the path fibration \(\Omega M \to LM \to M\) and he obtains sufficient and necessary conditions that \(\Omega M\) is totally non-homologous to zero in \(LM\) with respect to \(\mathbb Z/p\). His main method of proof is based on the Eilenberg-Moore spectral sequence and the Serre spectral sequence.