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 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.