Homotopy type of Frobenius complexes II

Abstract

A finitely generated additive submonoid $\Lambda$ of ${\mathbb{N}}^d$ has the partial order defined by $\lambda \le \lambda +\mu$ for $\lambda ,\mu \in \Lambda$. The Frobenius complex is the order complex of an open interval of $\Lambda$. In this paper, we express the homotopy type of the Frobenius complex of $\Lambda [\rho /r]$ in terms of those of $\Lambda$, where $\Lambda [\rho /r]$ is the additive monoid which is added the $r$-th part of $\rho$ to $\Lambda$. As applications, we determine the homotopy type of the Frobenius complex of some submonoids of $\mathbb{N}$, for example, the submonoid generated by a finite geometric sequence. We also state an application to the multi-graded Poincaré series.