  We consider the following question: if a simplicial complex $\Delta$ has
$d$-homology, then does the corresponding $d$-cycle always induce cycles of
smaller dimension that are not boundaries in $\Delta$? We provide an answer to
this question in a fixed dimension. We use the breaking of homology to show the
subadditivity property for the maximal degrees of syzygies of monomial ideals
in a fixed homological degree.
