Let K be a finite complex with a subcomplex \(K_ 0\), X a connected CW complex. Suppose M is a smooth, compact, parallelizable m-manifold with a submanifold \(M_ 0\) such that \((K,K_ 0)\cong (M,M_ 0)\), (e.g. M a regular neighbourhood of \(K\subset {\mathbb{R}}^ m)\). We prove: the space of based maps \(K/K_ 0\to S^ mX\) stably splits into a bouquet of spaces \(D_ k\) which depend on M, \(M_ 0\) and X. This is proved using configuration spaces and generalizes well-known results of Snaith, Goodwillie, et al.