Complexity of PL manifolds

Complexity of PL manifolds

We extend Matveev’s complexity of 3-manifolds to PL compact manifolds of arbitrary dimension, and we study its properties. The complexity of a manifold is the minimum number of vertices in a simple spine. We study how this quantity changes under the most common topological operations (handle additions, finite coverings, drilling and surgery of spheres, products, connected sums) and its relation...

Cataloguing PL 4-Manifolds by Gem-Complexity

We describe an algorithm to subdivide automatically a given set of PL nmanifolds (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...

Catalogues of PL-manifolds and complexity estimations via crystallization theory

Crystallization theory is a graph-theoretical representation method for compact PL-manifolds of arbitrary dimension, with or without boundary, which makes use of a particular class of edge-coloured graphs, which are dual to coloured (pseudo-) triangulations. These graphs are usually called gems, i.e. Graphs Encoding Manifolds, or crystallizations if the associated triangulation has the minimal ...

Non-pl Imbeddings of 3-manifolds

Metric Ricci curvature for $PL$ manifolds

We introduce a metric notion of Ricci curvature for PL manifolds and study its convergence properties. We also prove a fitting version of the Bonnet-Myers Theorem, for surfaces as well as for a large class of higher dimensional manifolds.