Gaps in the Milnor-Moore spectral sequence (Q1420736)

Ljusternik-Schnirelman category cat has the property \({\text{cat}}(X \cup e^n) \leq {\text{cat}}X+1\) where \(X\cup e^n\) is the space \(X\) with the attached cell \(e^n\). The authors prove that this property fails for certain well-known algebraic invariants that approximate cat (such as Toomer invariant \(e\), weak category, \(\sigma{\text{cat}}\)). For example, they construct a space \(Z\) such that \(e(Z)=2\) and \(e(Z\cup e^n)=4\). From these results, the authors deduce that there can be gaps in the Eilenberg-Moore spectral sequence, i.e. one can have \(E^{\infty}_{p,*}=0\) while \(E^{\infty}_{p+1,*}=0\).