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**1 - 9**of**9**### Cataloguing PL 4-manifolds by gem-complexity

"... We describe an algorithm to subdivide automatically a given set of PL n-manifolds (via coloured triangulations or, equivalently, via crystallizations) into classes whose elements are PL-homeomorphic. The algorithm, implemented in the case n = 4, succeeds to solve completely the PL-homeomorphism prob ..."

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We describe an algorithm to subdivide automatically a given set of PL n-manifolds (via coloured triangulations or, equivalently, via crystallizations) into classes whose elements are PL-homeomorphic. The algorithm, implemented in the case n = 4, succeeds to solve completely the PL-homeomorphism problem among the catalogue of all closed connected PL 4-manifolds up to gem-complexity 8 (i.e., which admit a coloured triangulation with at most 18 4-simplices). Possible interactions with the (not completely known) relationship among the different classifications in the TOP and DIFF=PL categories are also investigated. As a first consequence of the above PL classification, the non-existence of exotic PL 4-manifolds up to gem-complexity 8 is proved. Further applications of the tool are described, related to possible PL-recognition of different triangulations of the K3-surface.

### Computing Persistent Homology via Discrete Morse Theory

, 2013

"... This report provides theoretical justification for the use of discrete Morse the-ory for the computation of homology and persistent homology, an overview of the state of the art for the computation of discrete Morse matchings and motivation for an interest in these computations, particularly from th ..."

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This report provides theoretical justification for the use of discrete Morse the-ory for the computation of homology and persistent homology, an overview of the state of the art for the computation of discrete Morse matchings and motivation for an interest in these computations, particularly from the point of view of topological data analysis. Additionally, a new simulated anneal-ing based method for computing discrete Morse matchings is presented. For several problem instances this outperforms the best known heuristics for the task. The computation of homology and persistent homology has become an important task in computational topology, with applications in fields such as topological data analysis, computer vision and materials science. Unfortu-nately computing homology is currently infeasible for large input complexes. Discrete Morse theory enables the preprocessing of homology computation by reducing the size of the input complexes. This is advantageous from a