Minimal Crystallizations of 3- and 4- Manifolds
Biplab Basak · 2015
A simplicial cell complex K is the face poset of a regular CW complex W such that the boundary complex of each cell is isomorphic to the boundary complex of a simplex of same dimension. If a topological space X is homeomorphic to W then we say that K is a pseudotriangulation of X. For d 1, a (d + 1)-colored graph is a graph = (V; E) with a proper edge coloring : E ! f0; : : : ; dg. Such a graph is called contracted if (V; E n 1(i)) is connected for each color A contracted graph = (V; E) with an edge coloring : E ! f0; : : : ; dg determines a d-dimensional simplicial cell complex K( ) whose vertices have one to one correspondence with the colors 0; : : : ; d and the facets (d-cells) have one to one correspondence with the vertices in V . If K( ) is a pseudotriangulation of a manifold M then ( ; ) is called a crystallization of M. In [71], Pezzana proved that every connected closed PL manifold admits a crystallization. This thesis addresses many important results of crystallization theory in combinatorial topology. The main contributions in this thesis are the followings. We have introduced the weight of a group which has a presentation with number of relations is at most the number of generators. We have shown that the number of vertices of any crystallization of a connected closed 3-manifold M is at least the weight of the fundamental group of M. This lower bound is sharp for the 3-manifolds RP3, L(3; 1), L(5; 2), S1 S1 S1, S2 S1, S2 S1 and S3=Q8, where Q8 is the quaternion group. Moreover, there is a unique such vertex minimal crystallization in each of these seven cases. We have also constructed crystallizations of L(kq 1; q) with 4(q + k 1) vertices for q 3, k 2 and L(kq +1; q) with 4(q + k) vertices for q 4, k 1. In [22], Casali and Cristofori found similar crystallizations of lens spaces. By a recent result of Swartz [76], our crystallizations of L(kq + 1; q) are vertex minimal when kq + 1 are even. In [47], Gagliardi found presentations of the fundamental group of a manifold M in terms of a crystallization of M. Our construction is the converse of this, namely, given a presentation of the fundamental group of a 3-manifold M, we have constructed a crystallization of M. These results are in Chapter 3. We have de ned the weight of the pair (hS j Ri; R) for a given presentation hS j R of a group, where the number of generators is equal to the number of relations. We present an algorithm to construct crystallizations of 3-manifolds whose fundamental group has a presentation with two generators and two relations. If the weight of (hS j Ri; R) is n then our algorithm constructs all the n-vertex crystallizations which yield (hS j Ri; R). As an application, we have constructed some new crystallization of 3-manifolds. We have generalized our algorithm for presentations with three generators and a certain class of relations. For m 3 and m n k 2, our generalized algorithm gives a 2(2m + 2n + 2k 6 + n2 + k2)-vertex crystallization of the closed connected orientable 3-manifold Mhm; n; ki having fundamental group hx1;…