  A $d$-hypertree on $[n]$ is a maximal acyclic $d$-dimensional simplicial
complex with full $(d-1)$-skeleton on the vertex set $[n]$. Alternatively, in
the language of algebraic topology, it is a minimal $d$-dimensional simplicial
complex $T$ (assuming full $(d-1)$-skeleton) such that
$\tilde{H}_{d-1}(T;\mathbb{F})=0$.
  The $d$-hypertrees are a basic object in combinatorial theory of simplicial
complexes. They have been studied; and yet, many of their structural aspects
remain poorly understood.
  In this paper we study the boundaries $\partial_d T$ of $d$-hypertrees, and
the fundamental $d$-cycles defined by them. Our findings include:
  1. A full characterization of $\partial_d T$ over $\mathbb{F}_2$ for $d \leq
2$, and some partial results for $d \geq 3$.
  2. Lower bounds on the maximum size of a largest simple $d$-cycle on $[n]$.
In particular, for $d=2$, we construct a {\em Hamiltonian $d$-cycle} $H$ on
$[n]$, i.e., a simple $d$-cycle of size ${{n-1} \choose d} + 1$. For $d\geq 3$,
we construct a simple $d$-cycle of size ${{n-1} \choose d} - O(n^{d-2})$.
  3. Observing that the maximum of the expected distance between two vertices
chosen uniformly at random in a tree ($1$-hypertree) on $[n]$ is at most
$\thicksim n/3$, attained on Hamiltonian paths, we ask a similar question about
$d$-hypertrees. "How large can be the {\em average} size of a fundamental cycle
of a $d$-hypertree $T$ (i.e., the expected size of the dependency created by
adding a $d$-simplex on $[n]$, chosen uniformly at random, to $T$)?" For every
$d \in \mathbb{N}$, we construct an infinite family of $d$-hypertrees $\{T\}$
with the average size of a fundamental cycle at least $c_d\, |T| \,=\,
c_d\,{n-1 \choose d}$, where $c_d$ is a constant depending on the dimension $d$
alone.
