  A symmetry broken phase of a system with internal degrees of freedom often
features a complex order parameter, which generates a rich variety of
topological excitations and imposes topological constraints on their
interaction (topological influence); yet the very complexity of the order
parameter makes it difficult to treat topological excitations and topological
influence systematically. To overcome this problem, we develop a general method
to calculate homotopy groups and derive decomposition formulas which express
homotopy groups of the order parameter manifold $G/H$ in terms of those of the
symmetry $G$ of a system and those of the remaining symmetry $H$ of the state.
By applying these formulas to general monopoles and three-dimensional
skyrmions, we show that their textures are obtained through substitution of the
corresponding $\mathfrak{su}(2)$-subalgebra for the $\mathfrak{su}(2)$-spin. We
also show that a discrete symmetry of $H$ is necessary for the presence of
topological influence and find topological influence on a skyrmion
characterized by a non-Abelian permutation group of three elements in the
ground state of an SU(3)-Heisenberg model.
