Low-dimensional (co)homology of free Burnside monoids
Let \(r\geq 0\), \(m,d\geq 1\) be integers and let \(M_{r,m,d}\) be the free Burnside monoid defined by the presentation \({\mathcal B}_{r,m,d}=\langle\Sigma\mid X^{m+d}=X^m\), \(X\) any word \(\rangle\), where \(\Sigma\) is a set of cardinality \(r\). In the article, the second and third integral (co)homology groups of \(M_{r,m,d}\) are calculated: it is proved that for \(r>1\), \(m\geq 3\), \(d\geq 1\), \(H_2(M_{r,m,d})\) is a free abelian group of infinite rank and that \(H_3(M_{r,m,d})\) is a direct sum of infinitely many copies of \(Z_d\).
scientific article; zbMATH DE number 897482
scientific article