Minimal Crystallizations of 3 and 4 Manifolds
Abstract
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 ddimensional simplicial cell complex K( ) whose vertices have one to one correspondence with the colors 0; : : : ; d and the facets (dcells) 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 3manifold M is at least the weight of the fundamental group of M. This lower bound is sharp for the 3manifolds 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 3manifold 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 3manifolds 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 nvertex crystallizations which yield (hS j Ri; R). As an application, we have constructed some new crystallization of 3manifolds.
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 3manifold Mhm; n; ki having fundamental group hx1; x2; x3 j xm1 = xn2 = xk3 = x1x2x3i. These crystallizations are minimal and unique with respect to the given presentations. If `n = 2' or `k 3 and m 4' then our crystallization of Mhm; n; ki is vertexminimal for
all the known cases. These results are in Chapter 4.
We have constructed a minimal crystallization of the standard PL K3 surface. The corresponding simplicial cell complex has face vector (5; 10; 230; 335; 134). In combination with known results, this yields minimal crystallizations of all simply connected PL 4manifolds of \standard" type, i.e., all connected sums of CP2, CP2, S2 S2, and the K3 surface. In particular, we obtain minimal crystallizations of a pair 4manifolds which are homeomorphic but not PLhomeomorphic. We have also presented an elementary proof of the uniqueness of the 8vertex crystallization of CP2. These results are in Chapter 5.
For any crystallization ( ; ) the number f1(K( )) of 1simplices in K( ) is at least
d+1 . It is easy to see that f1(K( )) = d+1 if and only if (V; 1(A)) is connected for each d 2 2 1)set A called simple. All the crystallization in Chapter 5 (. Such a crystallization is are simple. Let ( ; ) be a crystallization of M, where = (V; E) and : E ! f0; : : : ; dg. We say that ( ; ) is semisimple if (V; 1(A)) has m + 1 connected components for each (d 1)set A, where m is the rank of the fundamental group of M.
Let ( ; ) be a connected (d +1)regular (d +1)colored graph, where = (V; E) and
: E ! f0; : : : ; dg. An embedding i : ,! S of into a closed surface S is called regular if there exists a cyclic permutation ("0; "1; : : : ; "d) (of the color set) such that the boundary of each face of i( ) is a bicolor cycle with colors "j; "j+1 for some j (addition is modulo d+1). Then the regular genus of ( ; ) is the least genus (resp., half of genus) of the orientable (resp., nonorientable) surface into which embeds regularly. The regular genus of a closed connected PL 4manifold M is the minimum regular genus of its crystallizations.
For a closed connected PL 4manifold M, we have provided the following: (i) a lower bound for the regular genus of M and (ii) a lower bound of the number of vertices of any crystallization of M. We have proved that all PL 4manifolds admitting semisimple crystallizations, attain our bounds. We have also characterized the class of PL 4manifolds which admit semisimple crystallizations. These results are in Chapter 6.
Collections
 Mathematics (MA) [115]
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