Breaking up Simplicial Homology and Subadditivity of Syzygies

We consider the following question: if a simplicial complex $\mathrm{\Delta }$ has $d$-homology, then does the corresponding $d$-cycle always induce cycles of smaller dimension that are not boundaries in $\mathrm{\Delta }$? 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.