The depth and LS category of a topological space - Félix, Yves; Halperin, Steve (Mathematica Scandinavica, 2018)
The depth of an augmented ring ε A→ k is the least p, or ∞, such that equation* _A^p(k , A) eq 0. equation* When X is a simply connected finite type CW complex, H_*(Ω X;Q) is a Hopf algebra and the universal enveloping algebra of the Lie algebra L_X of primitive elements. It is known that H_*(Ω X;Q) ≤ X, the Lusternik-Schnirelmann category of X. For any connected CW complex we construct a completion H(Ω X) of H_*(Ω X;Q) as a complete Hopf algebra with primitive sub Lie algebra L_X, and define X to be the least p or ∞ such that \[ ^p_UL_X(Q, H(Ω X)) eq 0. \] Theorem: for any connected CW complex, X≤ X.