Abstract
The goal of this paper is to give some theorems which relate to the problem of classifying combinatorial (resp. smooth) closed 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 consider this invariant, and extend in this context the concept of average order first introduced by Luo and Stong in 1993, and successively investigated by Tamura in 1996 and 1998. Then we obtain some classification results for closed connected smooth low-dimensional manifolds according to reduced complexity and average order. Finally, we answer to a question posed by Trout in 2013.
Highlights
All spaces and maps will be considered in the PL category, for which we refer to [16]
The cellular complex K = K(G) associated to G is constructed as follows: (1) for each vertex v of G, consider a standard n–simplex σn(v), and label its n + 1 vertices by the colors of ∆n; (2) if v and w are joined in G by an i–colored edge, identify the (n − 1)–faces of σn(v) and σn(w) opposite the vertex labelled by i ∈ ∆n, so that labelled vertices coincide
In [8], it was given the classification of the closed connected PL 5– manifolds up to reduced complexity 20
Summary
All spaces and maps will be considered in the PL category, for which we refer to [16]. A crystallization of M is an (n + 1)–colored graph (G, c) representing M such that K(G) has exactly n + 1 vertices (which we shall always assume to be colored by the elements of ∆n). In this case, K(G) is called a contracted triangulation of M. Let M be a closed connected PL n–manifold, (G, c) a crystallization of M (with color set ∆n), and K = K(G) the associated contracted triangulation of M. An (n + 1)-colored graph (G, c) is a crystallization of a closed connected PL n-manifold if and only if every partial subgraph Gi is connected and represents the (n − 1)-sphere, for every i ∈ ∆n. We are interested in two of them, called reduced complexity and average order, which will be presented in the two sections together with new results about characterizations of certain PL manifolds, up to PL homeomorphisms
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