Homotopy Types of Frobenius Complexes

Abstract

Let Λ be a submonoid of the additive monoid ${\mathbb{N}}$. There is a natural order on Λ defined by ${\lambda \leq \lambda +\mu}$ for ${\lambda,\mu \in \Lambda}$. A Frobenius complex of Λ is defined to be the order complex of an open interval of Λ. Suppose ${r \geq 2}$ and let ${\rho}$ be a reducible element of Λ. We construct the additive monoid ${\Lambda[\rho/r]}$ obtained from Λ by adjoining a solution to the equation ${r\alpha=\rho}$. We show that any Frobenius complex of ${\Lambda[\rho/r]}$ is homotopy equivalent to a wedge of iterated suspensions of Frobenius complexes of Λ. As a consequence, we derive a formula for the multi-graded Poincaré series associated to ${\Lambda[\rho/r]}$. As an application, we determine the homotopy types of the Frobenius complexes of some additive monoids. For example, we show that if Λ is generated by a finite geometric sequence, then any Frobenius complex of Λ is homotopy equivalent to a wedge of spheres.