| Let R be a subring of Q containing 16 or R = Fp with p > 3. Let (A, d) be a differential graded algebra (dga) such that its homology, H(A, d), is R-free and is the universal enveloping algebra of some Lie algebra L0. Given a free R-module V and a map d : V → A such that there is an induced map d' : V → L 0, let B denote the canonical dga extension (A II TV , d) where TV is the tensor algebra generated by V. H B is studied in the case that the Lie ideal [d 'V] ⊂ L0 is a free Lie algebra. This is a broad generalization of the previously studied inert condition. An intermediate semi-inert condition is introduced under which the algebraic structure of H B is determined.; This problem is suggested by the following cell attachment problem , first studied by J. H. C. Whitehead around 1940. For a simply-connected finite-type CW-complex X, denote by Y the space obtained by attaching a finite-type wedge of cells to X. Then how is the loop space homology H*(O Y; R) related to H*(OX; R)? In some cases this topological problem is described by the previous algebraic situation. Under a further hypothesis it is shown that if H*(OX; R) is generated by Hurewicz images then so is H*(OY; R) and if R ⊂ Q then the localization of OY at R is homotopy equivalent to a product of spheres and loop spaces on spheres.; It is well-known that if the coefficient ring is a field, Lie subalgebras of free Lie algebras are also free. To help implement the above results this fact is generalized, giving a simple condition guaranteeing that Lie subalgebras of more general Lie algebras are free. |
