Remez, JJ; (2012) Diagonal resolutions for the metacyclic groups G(pq). Doctoral thesis , UCL (University College London).

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Abstract

We study the notion of a free resolution. In general a free resolution can be of any length depending on the group ring under investigation. We consider the metacyclic groups G(pq) which admit periodic resolutions. In such circumstances it is possible to achieve fully \emph{diagonalised resolutions}. By discussing the representation theory over integral group rings we obtain a complete list of indecomposable modules over Z[G(pq)]. Such a list aids the decomposition of the augmentation ideal (the first syzygy) into a direct sum of indecomposable modules. Therefore we are able to achieve a diagonalised map here. From this point it is possible to decompose all of the remaining syzygies in terms of indecomposable modules, leaving a diagonal resolution in principle. The existence of these diagonal resolutions significantly simplify a problem in low-dimensional topology, namely the R(2)-D(2) problem. There are two stages to verifying this problem, and we prove the first stage using cohomological properties of the syzygy decompositions. The second stage is realising the Swan map. Although we do not manage to realise it fully, we are able to realise certain terms. Finally this thesis includes an in depth exposition of the R(2)-D(2) for the non-abelian group of order 21. In this case a positive result has been achieved using an explicitly calculated diagonal resolution.

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