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An affine monoid is an additive monoid which is cancellative, pointed andfinitely generated. An affine monoid $\Lambda$ has the partial order defined by$\lambda \le \lambda + \mu$. The Frobenius complex is the order complex of anopen interval of $\Lambda$ with respect to this partial order. The reducedhomology of the Frobenius complex is related to the torsion group of the monoidalgebra $K[\Lambda]$. In this paper, we pay attention to homotopy types ofFrobenius complexes, and we express the homotopy types of the Frobeniuscomplexes of $\Lambda$ in terms of those of $\Lambda_1$ and $\Lambda_2$ when$\Lambda$ is an affine monoid obtained by gluing two affine monoids $\Lambda_1$and $\Lambda_2$ with one relation. We also state an application to thePoincar\'e series of the torsion group of the monoid algebra.

Frobenius complexes and the homotopy colimit of a diagram of posets over a poset