W. H. Mannan

Realizing algebraic 2-complexes by cell complexes

The realization theorem asserts that for a finitely presented group G, the D(2) property and the realization property are equivalent as long as G satisfies a certain finiteness condition. We show that the two properties are in fact equivalent for all finitely presented groups

Computing Borel's Regulator

We present an infinite series formula based on the Karoubi-Hamida integral, for the universal Borel class evaluated on H_{2n+1}(GL(C)). For a cyclotomic field F we define a canonical set of elements in K_3(F) and present a novel approach (based on a free differential calculus) to constructing them. Indeed, we are able to explicitly construct their images in H_3(GL(C)) under the Hurewicz map. Applying our formula to these images yields a value V_1(F), which coincides with the Borel regulator R1(F) when our set is a basis of K_3(F) modulo torsion. For F = Q(e^{2?i/3}) a computation of V_1(F) has been made based on our techniques.

Koszul-Morita duality

We construct a generalization of Koszul duality in the sense of Keller--Lef\`evre for not necessarily augmented algebras. This duality is closely related to classical Morita duality and specializes to it in certain cases.

Explicit generators for the relation module in the example of Gruenberg–Linnell

Gruenberg and Linnell showed that the standard relation module of a free product of n groups of the form C r ×

Quillen’s Plus Construction and the D(2) problem

\mathbb{Z}

Homotopy types of truncated projective resolutions

could be generated by just n + 1 generators, raising the possibility of a relation gap. We explicitly give such a set of generators.