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On some homotopical and homological properties of monoid presentations

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Abstract

If a monoid S is given by some finite complete presentation ℘, we construct inductively a chain of CW-complexes

$$\mathcal{D}\subset \Delta_{2}\subset(\mathcal{D},\mathbf{p}_{1})\subset\Delta_{3}\subset \cdots\subset \Delta_{n-1}\subset (\mathcal{D},\mathbf{p}_{1},\ldots,\mathbf{p}_{n-2})\subset \Delta_{n},$$

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 (ℤS,ℤS)-bimodules

$$0\rightarrow H_{m}(\mathcal{D},\mathbf{p}_{1},\ldots,\mathbf{p}_{m-1})\stackrel{\Phi }{\longrightarrow}\mathbb{Z}S.\mathbf{p}_{m-1}.\mathbb{Z}S\stackrel{\nu}{\longrightarrow }H_{m-1}(\mathcal{D},\mathbf{p}_{1},\ldots,\mathbf{p}_{m-2})\rightarrow 0,$$

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 ℤS. Also we show that for groups properties FDT and FHT coincide.

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  1. Fakulteti i Shkencave të Natyrës, Departamenti i Matematikës, Universiteti i Tiranës, Tiranë, Albania

    Elton Pasku

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  1. Elton Pasku
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Correspondence to Elton Pasku.

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Communicated by Steve Pride.

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Pasku, E. On some homotopical and homological properties of monoid presentations. Semigroup Forum 76, 427–468 (2008). https://doi.org/10.1007/s00233-007-9037-1

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