Cell attachments and the homology of loop spaces and differential graded algebras

Abstract

The cell-attachment problem, perhaps first studied by J.H.C. Whitehead around 1940, asks one to describe the effect of attaching one or more cells, on the algebraic invariants of a topological space. This thesis studies the effect of cell attachments on loop space homology. It generalizes previous work by D. Anick on spherical 2-cones, and S. Halperin, J.-M. Lemaire and others on inert attaching maps. It is assumed that the loop space homology of the base space is torsion-free. Two classes of attaching maps are introduced: free and semi-inert attaching maps. For these cases, the loop space homology is calculated as a module and as an algebra, respectively, and is shown to have a surprisingly simple form. These results are also presented in terms of extensions of differential graded algebras. The second part of the thesis uses Adams-Hilton models and the previous results to prove that the loop space homology of certain CW complexes is generated by Hurewicz images. This can sometimes be used to show that the loop space of a finite complex is, after localizing away from finitely many primes, homotopy equivalent to a product of spheres and loops on spheres. A lemma, which may of independent interest, generalizes the Schreier property, which states that a Lie subalgebra of a free (graded) Lie algebra is also a free Lie algebra.