#### PhD Thesis colloquium

##### Venue: Lecture Hall III, Department of Mathematics

This thesis addresses many important results of crystallization theory in combinatorial topology. The main contributions in this thesis are the followings:

(i) We introduce the weight of a group which has a presentation with number of relations is at most the number of generators. We prove 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 $\mathbb{R P}^3$, $L(3,1)$, $L(5,2)$, $S^1\times S^1 \times S^1$, $S^{\hspace{.2mm}2} \times S^1$, $\TPSS$ and $S^{\hspace{.2mm}3}/Q_8$, where $Q_8$ is the quaternion group. Moreover, there is a unique such vertex minimal crystallization in each of these seven cases. We also construct crystallizations of $L(kq-1,q)$ with $4(q+k-1)$ vertices for $q \geq 3$, $k \geq 2$ and $L(kq+1,q)$ with $4(q+k)$ vertices for $q\geq 4$, $k\geq 1$. By a recent result of Swartz, our crystallizations of $L(kq+1, q)$ are facet minimal when $kq+1$ are even.

(ii) We present an algorithm to find certain types of crystallizations of $3$-manifolds from a given presentation $\langle S \mid R \rangle$ with $\#S=\#R=2$. We generalize this algorithm for presentations with three generators and certain class of relations. This gives us crystallizations of closed connected orientable 3-manifolds having fundamental groups $\langle x_1,x_2,x_3 \mid x_1^m=x_2^n=x_3^k=x_1x_2x_3 \rangle$ with $4(m+n+k-3)+ 2\delta_n^2 + 2 \delta_k^2$ vertices for $m\geq 3$ and $m \geq n \geq k \geq 2$, where $\delta_i^j$ is the Kronecker delta. If $n=2$ or $k\geq 3$ and $m \geq 4$ then these crystallizations are vertex-minimal for all the known cases.

(iii) We found a minimal crystallization of the standard pl K3 surface. This minimal crystallization is a ‘simple crystallization’. Using this, we present minimal crystallizations of all simply connected pl $4$-manifolds of “standard” type, i.e., all the connected sums of $\mathbb{CP}^2$, $S^2 \times S^2$, and the K3 surface. In particular, we found minimal crystallizations of a pair of 4-manifolds which are homeomorphic but not pl-homeomorphic.

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Last updated: 05 Dec 2019