The goal of this paper is to give some theorems which relate to the problem of classifying smooth 4-manifolds up to piecewise -linear (PL) homeomorphism. For this, we use the combinatorial approach to the topology of PL manifolds by means of a special kind of edge-colored graphs, called crystallizations. Within this representation theory, Bracho and Montejano introduced in 1987 a nonnegative numerical invariant, called the reduced complexity, for any closed n-dimensional PL manifold. Here we obtain the complete classification of all closed connected smooth 4-manifolds of reduced complexity less than or equal to 14.
Classifying combinatorial 4-manifolds up complexity / Cavicchioli, A.; Spaggiari, F.. - In: BOLETÍN DE LA SOCIEDAD MATEMÁTICA MEXICANA. - ISSN 1405-213X. - 14:2(2008), pp. 303-319.
Classifying combinatorial 4-manifolds up complexity
Cavicchioli A.;Spaggiari F.
2008
Abstract
The goal of this paper is to give some theorems which relate to the problem of classifying smooth 4-manifolds up to piecewise -linear (PL) homeomorphism. For this, we use the combinatorial approach to the topology of PL manifolds by means of a special kind of edge-colored graphs, called crystallizations. Within this representation theory, Bracho and Montejano introduced in 1987 a nonnegative numerical invariant, called the reduced complexity, for any closed n-dimensional PL manifold. Here we obtain the complete classification of all closed connected smooth 4-manifolds of reduced complexity less than or equal to 14.
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