Ph.D Thesis | |

Ph.D Student | Wallach Nathan Shimon |
---|---|

Subject | Topological and Combinatorial Properties of Random Simplicial Complexes |

Department | Department of Mathematics |

Supervisor | PROF. Roy Meshulam |

This research considers *d*-dimensional
simplicial complexes which have a complete (*d*-1)-skeleton. We define the
model *K*(*n,d,p*) of random complexes of this type on *n*
vertices. In this model, we study the (*d-*1)-dimensional homological
connectivity of a random complex. We determine that the probability *p* =
(*d* log *n*) / *n* is the threshold probability at which the
homology group *H _{d}*

Our approach builds upon recent
results of Linial and Meshulam, who showed such a result in two dimensions for
the homology group *H*_{1}(*K*,*Z*_{2}). We
extend this to all dimensions and to homology groups with other coefficient
groups.

The model we consider is quite
general. In the two dimensional case we show how to construct a subcomplete
complex whose fundamental group is any desired finitely presented group. A
consequence of this result is that in the two-dimensional case, for a
sufficiently large value of *n*, the fundamental group (the homology group
*H*_{1}), can be any possible finitely presented group (finitely
generated abelian group) at the time when the last edge (1-simplex) is covered
in the complex process model.

We also determine the minimal
degree of a (*d*-1) -simplex which suffices to guarantee that the complex
has a trivial (*d*-1)-dimensional homology group. This generalizes a
classic result of this nature for graphs.

In the last chapter of this
work, we give an elementary approach to the Steinberg representation of the
general linear group *GL _{n}*(