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Computer search for nilpotent complexes. (English) Zbl 0898.55010

The authors, together with a team of undergraduate students at Fordham University, used computers to search for three-dimensional finite nilpotent complexes over groups of the form \(\mathbb{Z}_n\oplus \mathbb{Z}_m\). Such complexes were eventually found for \(\mathbb{Z}_2\oplus \mathbb{Z}_6\), \(\mathbb{Z}_2\oplus \mathbb{Z}_{10}\), and \(\mathbb{Z}_3\oplus \mathbb{Z}_6\). The present paper describes the strategy for constructing nilpotent complexes of dimension three, and some of the issues in implementing the computer search. The main computational issues are “normalizing” matrices, especially to the Smith normal form, and mapping matrices over \(\mathbb{Z}\) to matrices over \(\mathbb{Z}_p\) for various primes \(p\). The authors conclude with a summary of the complexes discovered and open questions.

MSC:

55P99 Homotopy theory

Software:

Fermat
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References:

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