L-S category and homogeneous spaces (Q914190)

The author gives some relations between rational Lusternik-Schnirelmann type invariants for homogeneous spaces. For a 1-connected space S, the Milnor-Moore spectral sequence is obtained by filtering the bar construction on the chain differential algebra \(C_*(\Omega S)\) by the bar degree. Toomer and Ginsburg have used this spectral sequence to introduce two numerical invariants: \[ e(S)=\sup \{p| \quad E_{\infty}^{p,*}\neq 0\}\text{ and } \ell (S)=\inf \{p\geq 1| \quad E^{*,*}_{p+1}=E_{\infty}^{*,*}\}. \] With rational coefficients the corresponding invariants are respectively denoted by \(e_ 0| S)\) and \(\ell_ 0| S)\). If G and H are compact, simply connected Lie groups, the author proves that \(cat_ 0(G/H)\leq e_ 0(G_ H)\cdot \ell_{\theta}(G/H)\). He also proves the Halperin's conjecture \((``cat_ 0(G/H)=e_ 0(G/H)'')\) in some particular cases.