On some homotopical and homological properties of monoid presentations. (Q927282)

Extending the Squier complex, the author constructs a chain of CW-complexes. Let \(\mathbf{(x,r)}\) be a finite complete presentation of a monoid \(M\) and let \(\mathcal D\) be the Squier complex associated with \(\mathbf{(x,r)}\) [\textit{C. C. Squier, F. Otto} and \textit{Y. Kobayashi}, Theor. Comput. Sci. 131, No. 2, 271-294 (1994; Zbl 0863.68082)]. A 3-dimensional CW-complex \((\mathcal D,\mathbf p_1)\) is constructed by adding 2-cells \(\mathbf p_1\) which come from critical overlappings of edges in \(\mathcal D\). Then add 3-cells which come from overlappings of 2-cells and edges in \((\mathcal D,\mathbf p_1)\). The author generalizes this construction up to any dimension to get a chain of complexes \(\mathcal D\subset(\mathcal D,\mathbf p_1)\subset\cdots\subset(\mathcal D,\mathbf p_1,\dots,\mathbf p_{n-2})\). This construction gives another proof of the result that a monoid with finite complete presentation is of type \(\text{bi-FP}_\infty\) [\textit{Y. Kobayashi}, Trans. Am. Math. Soc. 357, No. 3, 1095-1124 (2005; Zbl 1069.16010)].