Abstract: The notion of a partial product space is a relatively recent unification of various combinatorial constructions in topology. This construction is variously known as the generalized moment-angle complex, or (more euphoniously) as the polyhedral product functor. Some instances of it are closely related to Davis and Januszkiewicz's quasitoric manifolds: these include the moment-angle complexes (Buchstaber and Panov) and homotopy orbit spaces for quasitoric manifolds. By making suitable choices, one also obtains classifying spaces for right-angled Artin groups and Coxeter groups, as well as certain real and complex subspace arrangements.

One advantage to this generality is that some topological information about such spaces can sometimes be expressed directly in combinatorial terms: presentations of cohomology rings; a homotopy-theoretic decomposition of the suspension of a partial product space; descriptions of rational homotopy Lie algebras and the Pontryagin algebra. I will give an introductory overview of some remarkable results along these lines.