On some homotopical and homological properties of monoid presentations.

If a monoid S is given by some finite complete presentation ℘, we construct inductively a chain of CW-complexes such that Δ n has dimension n, for every 2≤ m≤ n, the m-skeleton of Δ n is Δ m , and p m are critical ( m+1)-cells with 1≤ m≤ n−2. For every 2≤ m≤ n−1, the following is an exact sequence of (ℤ S,ℤ S)-bimodules where $(\mathcal{D},\mathbf{p}_{1},\ldots,\mathbf{p}_{m-2})=\mathcal{D}$ if m=2. We then use these sequences to obtain a free finitely generated bimodule partial resolution of ℤ S. Also we show that for groups properties FDT and FHT coincide. [ABSTRACT FROM AUTHOR]