  Cheeger-type inequalities in which the decomposability of a graph and the
spectral gap of its Laplacian mutually control each other play an important
role in graph theory and network analysis, in particular in the context of
expander theory. The natural problem, however, to extend such inequalities to
simplicial complexes and their higher order Eckmann Laplacians has been open
for a long time. The question is not only to prove an inequality, but also to
identify the right Cheeger-type constant in the first place. Here, we solve
this problem. Our solution involves and combines constructions from simplicial
topology, signed graphs, Gromov filling radii and an interpolation between the
standard 2-Laplacians and the analytically more difficult 1-Laplacians, for
which, however, the inequalities become equalities. It is then natural to
develop a general theory for $p$-Laplacians on simplicial complexes and
investigate the related Cheeger-type inequalities.
