On the cohomology of homogeneous spaces of finite loop spaces and the Eilenberg-Moore spectral sequence
The author generalizes \textit{A. Borel}'s theorem [Ann. Math. (2) 57, 115-207 (1953; Zbl 0052.40001)] from Lie groups to finite loop spaces. He shows that if \(p\) is prime, \(X\) is simply connected finite loop space which satisfies Lin's condition \({\mathcal Q}_p\), \(Y\) is a subgroup of \(X\) of maximal rank whose integral cohomology has no \(p\)-torsion and \(F\) is a field of characteristic \(p\), then \[ H^*(X/Y;F)\cong F\otimes_{H^*(B\widetilde X;F)}H^* (BY \times L;F). \] This implies the following generalization of \textit{R. Bott}'s theorem [Bull. Soc. Math. France 84, 251-281 (1956; Zbl 0073.40001)] from Lie groups to finite loop spaces. Let \(X\) be a simply connected finite loop space with maximal torus \(T\) such that \(X\) satisfies Lin's condition \({\mathcal Q}_p\) for every prime \(p\). Then \(H^*(X/T;Z)\) is zero in odd degrees and is torsion-free in even degrees. These theorems are derived from the following pullback fibration. \[ \begin{matrix} X/Y & = & X/Y\\ \downarrow & & \downarrow \\ BY\times L & \to & BY\\ \downarrow & & \downarrow\\ B \widetilde X & \to & BX \end{matrix} \] Here \(L\) is a product of spaces of the \(\Omega\)-spectrum associated to the Johnson-Wilson space \(BB\langle 1\rangle\), and \(\widetilde X\) is the homotopy fibre of a map \(X\to L\). The author shows that the Eilenberg-Moore spectral sequence of the pullback fibration collapses using a structure theorem for \(\text{Tor}_A(F,B)\) in the case where \(f:A\to B\) is an algebra homomorphism of polynomial algebras which makes \(B\) a coherent \(A\)-module.
scientific article; zbMATH DE number 1340424