  The fundamental group of the $2$-dimensional Linial-Meshulam random
simplicial complex $Y_2(n,p)$ was first studied by Babson, Hoffman and Kahle.
They proved that the threshold probability for simple connectivity of
$Y_2(n,p)$ is about $p\approx n^{-1/2}$. In this paper, we show that this
threshold probability is at most $p\le (\gamma n)^{-1/2}$, where $\gamma =
4^4/3^3$, and conjecture that this threshold is sharp.
  In fact, we show that $p=(\gamma n)^{-1/2}$ is a sharp threshold probability
for the stronger property that every cycle of length $3$ is the boundary of a
subcomplex of $Y_2(n,p)$ that is homeomorphic to a disk. Our proof uses the
Poisson paradigm, and relies on a classical result of Tutte on the enumeration
of planar triangulations.
