Edwards, T.; (2006) Algebraic 2-complexes over certain infinite abelian groups. Doctoral thesis , University of London.

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Abstract

Whitehead's Theorem allows the study of homotopy types of two dimensional CW complexes to be phrased in terms of chain homotopy types of algebraic complexes, arising as the cellular chains on the universal cover. It is natural to ask whether the category of algebraic complexes fully represents the category of CW complexes, in particular whether every algebraic complex is realised geometrically. The case of two dimensional complexes is of special interest, partly due to the relationship between such complexes and group presentations and partly since, as was recently proved, it relates to the question as to when cohomology is a suitable indicator of dimension. This thesis has two primary considerations. The first is the generalisation to infinite groups of F.E.A. Johnson's approach regarding problems of geometric realisation. It is shown, under certain restrictions, that the class of projective extensions containing algebraic complexes may be recognised as the unit elements of a ring, with ring elements congruence classes of extensions of the trivial module by a second homotopy module. The realisation property is shown to hold for the free abelian groups on two and three generators, and for the product of a cyclic group and a free group on a single generator. Secondly, a reinterpretation is given of the well documented relation ship between the congruence classes represented by Swan modules and the projective modules constructed via Milnor's connecting homomorphism and the relevant fibre product diagram. This relationship is shown to be typical of projective modules occurring in extensions of a two-sided ideal by a quotient ring, and we show that any two-sided ideal in a general ring results in a Mayer-Vietoris sequence which is different and complimentary to the standard excision sequence.

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