A cellular topological field theory

We present a construction of cellular BF theory (in both abelian and non-abelian variants) on cobordisms equipped with cellular decompositions. Partition functions of this theory are invariant under subdivisions, satisfy a version of the quantum master equation, and satisfy Atiyah-Segal-type gluing formula with respect to composition of cobordisms

First steps in brave new commutative algebra

This is an expository account of completion and local cohomology in brave new commutative algebra, especially as it applies to completion theorems and their duals in equivariant topology. The first part is fairly direct, but the second considers Morita theory and Gorenstein ring spectra. This draws on work of the author with Benson, Dwyer, Iyengar and May, amongst others

Moment-angle complexes, monomial ideals, and Massey products

Associated to every finite simplicial complex K there is a "moment-angle" finite CW-complex, Z_K; if K is a triangulation of a sphere, Z_K is a smooth, compact manifold. Building on work of Buchstaber, Panov, and Baskakov, we study the cohomology ring, the homotopy groups, and the triple Massey products of a moment-angle complex, relating these topological invariants to the algebraic combinatorics of the underlying simplicial complex. Applications to the study of non-formal manifolds and subspace arrangements are given.Comment: 30 pages. Published versio

Duality in algebra and topology

In this paper we take some classical ideas from commutative algebra, mostly ideas involving duality, and apply them in algebraic topology. To accomplish this we interpret properties of ordinary commutative rings in such a way that they can be extended to the more general rings that come up in homotopy theory. Amongst the rings we work with are the differential graded ring of cochains on a space, the differential graded ring of chains on the loop space, and various ring spectra, e.g., the Spanier-Whitehead duals of finite spectra or chromatic localizations of the sphere spectrum. Maybe the most important contribution of this paper is the conceptual framework, which allows us to view all of the following dualities: Poincare duality for manifolds, Gorenstein duality for commutative rings, Benson-Carlson duality for cohomology rings of finite groups, Poincare duality for groups, Gross-Hopkins duality in chromatic stable homotopy theory, as examples of a single phenomenon. Beyond setting up this framework, though, we prove some new results, both in algebra and topology, and give new proofs of a number of old results