Combinatorial embeddings of manifolds

The following results on embedding manifolds resemble in their form Dehn's Lemma, the Sphere Theorem, and, especially, embedding theorems obtained for differentiable manifolds by A. Haefliger [i]. Let M, Q be finite combinatorial manifolds of dimensions m and q, respectively. Let M, Q be their boundaries (possibly empty), and let ƒ : M—>Q be a piecewise linear map. We define sing (ƒ) to be the closure in M of the set {xÇ,M\ f~f(x)^x). Let R = MC\f-(Q), S be a regular neighbourhood of R in M (see [3]) and T=M — S. THEOREM 1. Of the following conditions, (i), (ii), (iii), and any one of (iv), (v), (vi) are sufficient to ensure the existence of a piecewise linear embedding g: MQQ such that g is homotopic to f rel. M: (i) q^m+3, (ii) M is (2m —q) connected, (iii) Q is (2m — q+1) connected, (iv) /(M)CO.. (v) sing(f)C\M — 0 and T is (3m — 2q+l) connected, (vi) sing(f)C\R = 0 and T is (2m — q — l) connected.