Generation of Catalogues of PL n-manifolds: Computational Aspects on HPC Systems

Abstract

Abstract. Within mathematical research, Geometric Topology deals with the study of piecewise-linear n-manifolds, i.e. triangulable spaces which appear locally as the n-dimensional Euclidean space. This paper reports on the computational aspects of an algorithm for generating triangulations of PL 3-and 4-manifolds represented by edge-coloured graphs. As the number of graph vertices is increased the algorithm becomes computationally expensive very quickly, making it a natural candidate for the usage of HPC resources. We present an optimized, parallel version of the algorithm that is suitable for deployment of multi-core systems. Scalability results are discussed on two different platforms, namely an IBM iDataPlex Linux cluster and the IBM supercomputer BlueGene/Q.Key words: High Performance Computing, n-manifolds, coloured triangulations, edge-coloured graphs.AMS subject classifications. 57Q15 -57M15 -68W10.1. Introduction. Geometric Topology deals with piecewise-linear (PL) n-manifolds [2], i.e. compact topological manifolds for which there is a triangulation such that each point has a neighbourhood which is piecewise-linearly isomorphic (PL-homeomorphic) to an affine n-simplex. Catalogues of triangulations of PL n-manifolds are valuable sources of data: they yield examples to test conjectures, calculate invariants and make comparisons; they can offer insight into the structural properties of the represented manifolds and may suggest ideas for further theoretical investigation.In particular, since each compact (topological) 3-manifold admits a PL structure and any two PL structures on the same topological 3-manifold are equivalent (i.e. PL-homeomorphic, see [2]), the study of triangulations of PL 3-manifolds is naturally related to the problem of classification, which is still one of the main topics of 3-dimensional topology. The possibility of representing manifolds by combinatorial structures, together with recent advances in computing power, enabled topologists to construct exhaustive tables of small (i.e. obtained by a small number of simplices) 3-manifolds based on different representation methods. In the closed case (i.e. compact and without boundary), catalogues have already been produced and analysed by many authors [8,19,20], with a particular focus on combinatorial properties of minimal triangulations.On the other hand, the problem of classification in dimension four must take into account that a topological 4-manifold not always admits PL structures or may admit non-equivalent ones. For example, although there exists a classification of simply-connected topological 4-manifolds, long established by Freedman [7], the study of (PL) equivalence classes of such structures, especially with regard to their minimal representatives, is an interesting and still open subject of research. Several examples of different PL 4-manifolds triangulating the same topological 4-manifold have recently been presented and the subject is being continuously updated (see for example [9]), but no exhaustive catalogue is available...

Cataloguing PL 4-Manifolds by Gem-Complexity

“…The above described restrictions succeed in reducing considerably both the computation time and the size of the resulting catalogues. Moreover, a parallelization strategy, which has been adopted in the implementation, has allowed to reduce further the computation time: see [41] for details.…”