It is shown that for $A_{k, m}$ a k-homogeneous $C^*$-algebra over $S^{2n - 1} imes S^1$ such that no non-trivial matrix algebra can be factored out of $A_{k, m}$ and $A_{k, m} otimes M_l(C)$ has a non-trivial bundle structure for any positive integer l, we construct an $A_{k, m^-} C(S^{2n - 1} imes S^1) otimes M_k(C)$-equivalence bimodule to show that every k-homogeneous $C^*$-algebra over $S^{2n - 1} imes S^1)$. Moreover, we prove that the tensor product of the k-homogeneous $C^*$-algebra $A_{k, m}$ with a UHF-algebra of type $p^infty$ has the tribial bundle structure if and only if the set of prime factors of k is a subset of the set of prime factors of pp.