Steve Halperin

CHAPTER 16 – Differential Graded Algebras in Topology

This chapter discusses the differential graded algebras in topology. The homotopy theory of topological spaces attempts to classify weak homotopy types of spaces and homotopy classes of maps. For a given geometric problem, however, it is often sufficient to replace the homotopy theory of spaces by a weaker “algebraic” homotopy theory as described, for instance, in Baues book. This chapter considers the specific case when spaces are modeled by the homotopy theory of differential graded algebras.

The rational LS category of products and of Poincare duality complexes

We give a new characterisation of rational LS category, cat(o)(X), from which two theorems follow. Theorem 1: cat(o)(X + Y) = cat(o)(X) + cat(o)(Y). Theorem 3: If X is a Poincare: duality complex, then cat(o)(X) = e(o)(X), where e(o) is Toomers invariant