  Homology features of spaces which appear in applications, for instance 3D
meshes, are among the most important topological properties of these objects.
Given a non-trivial cycle in a homology class, we consider the problem of
computing a representative in that homology class which is optimal. We study
two measures of optimality, namely, the lexicographic order of cycles (the
lex-optimal cycle) and the bottleneck norm (a bottleneck-optimal cycle). We
give a simple algorithm for computing the lex-optimal cycle for a 1-homology
lass in a closed orientable surface. In contrast to this, our main result is
that, in the case of 3-Manifolds of size $n^2$ in the Euclidean 3-space, the
problem of finding a bottleneck optimal cycle cannot be solved more efficiently
than solving a system of linear equations with an $n \times n$ sparse matrix.
From this reduction, we deduce several hardness results. Most notably, we show
that for 3-manifolds given as a subset of the 3-space of size $n^2$, persistent
homology computations are at least as hard as rank computation (for sparse
matrices) while ordinary homology computations can be done in $O(n^2 \log n)$
time. This is the first such distinction between these two computations.
Moreover, it follows that the same disparity exists between the height
persistent homology computation and general sub-level set persistent homology
computation for simplicial complexes in the 3-space.
