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.

- All seminars.
- Seminars for 2015

Last updated: 11 Dec 2023