  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 Poincar\'e series.
