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Suppose that each point in the decomposition space has arbitrarily small neighborhoods with ($n-1$)-sphere frontiers or boundaries which miss $\\pi(H_G)$. If all the arcs are tame in the particular area on the boundary of an $n$-cell $C$ in $S^n$, then this paper shows that this condition implies $S^n/G$ is homeomorphic to $S^n$ ($n\\geq 4$). This answers a weak form of a conjecture asked by Daverman [2, p. 61]. 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