The DGL N[sup(kl)]

We examine in Chapter 5 a sub-dgL N kl of M k II M l. Topologically, N kl corresponds to Σ(ΩΖ}*) Λ (ΩΖ}/). We begin with some general discussion of Hopf commutators and Hopf algebra kernels. We explore the obstructions to a free dgL having a “linear differential” when its enveloping algebra has one. Specializing to the case of N kl, we construct an isomorphism gu : C(Vkl) —> N kl, where V kl = (£/M*)+ ® (£/Mz)+, that enables us to access explicitly the elements of N kl. The main result of Chapter 5 is that gu is an isomorphism of dgL’s, and hence that (N kl,d ) has a “linear differential.” This fact is the algebraic analog of Σ(ΩΖ}*) Λ (ΩΖ}/) being a wedge of Moore spaces, which will be proved in Chapter 12.