  Topological excitations are usually classified by the $n$th homotopy group
$\pi_n$. However, for topological excitations that coexist with vortices, there
are case in which an element of $\pi_n$ cannot properly describe the charge of
a topological excitation due to the influence of the vortices. This is because
an element of $\pi_n$ corresponding to the charge of a topological excitation
may change when the topological excitation circumnavigates a vortex. This
phenomenon is referred to as the action of $\pi_1$ on $\pi_n$. In this paper,
we show that topological excitations coexisting with vortices are classified by
the Abe homotopy group $\kappa_n$. The $n$th Abe homotopy group $\kappa_n$ is
defined as a semi-direct product of $\pi_1$ and $\pi_n$. In this framework, the
action of $\pi_1$ on $\pi_n$ is understood as originating from noncommutativity
between $\pi_1$ and $\pi_n$. We show that a physical charge of a topological
excitation can be described in terms of the conjugacy class of the Abe homotopy
group. Moreover, the Abe homotopy group naturally describes vortex-pair
creation and annihilation processes, which also influence topological
excitations. We calculate the influence of vortices on topological excitations
for the case in which the order parameter manifold is $S^n/K$, where $S^n$ is
an $n$-dimensional sphere and $K$ is a discrete subgroup of $SO(n+1)$. We show
that the influence of vortices on a topological excitation exists only if $n$
is even and $K$ includes a nontrivial element of $O(n)/SO(n)$.
