Homotopy Types of Frobenius Complexes

Let Λ be a submonoid of the additive monoid N . There is a natural order on Λ defined by λ ≤ λ + μ for λ , μ ∈ Λ . A Frobenius complex of Λ is defined to be the order complex of an open interval of Λ. Suppose r ≥ 2 and let ρ be a reducible element of Λ. We construct the additive monoid Λ [ ρ / r ] obtained from Λ by adjoining a solution to the equation r α = ρ . We show that any Frobenius complex of Λ [ ρ / 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 Λ [ ρ / 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.