### Piecewise linear embeddings and isotopies

J. F. P. Hudson
1966 Bulletin of the American Mathematical Society
Let M and Q be PL manifolds of dimensions m and q respectively, M being compact. Let dM and dQ be their boundaries, possibly empty. If ƒ : M->Q is any continuous mapping, ir f (ƒ) will denote the relative homotopy group of the pair (C/, AT), Cf being the mapping cylinder of ƒ : M-+Q. THE EMBEDDING THEOREM. Let ƒ: M-*Q be a mapping such that f~l(dQ) =dM, and the restriction f \d M is a PL embedding. If q -'M è 3, *>(ƒ) = 0 for r ^ 2m -q + 1, TT r (M) = 0 forrS3m~2q+2 9 then f is homotopic,
more » ... g d M fixed f to a PL embedding. THE UNKNOTTING THEOREM. Let ƒ, g:M->Q be /wo PL embeddings withf^dQ -g^dQ -dM and f and g homotopic keeping d M fixed. If q -m ^ 3, *>(ƒ) -0 /or r S 2m -g + 2 and 7Tr(M) = 0 for r £ 3m -2q + 3, then f and g are PL ambient isotopic keeping dQ fixed. These two theorems include the combinatorial analogues of Haefliger's Embedding and Isotopy Theorems of [l] for differential manifolds, as well as Zeeman's Unknotting Theorem [ô] and the relevant form of Irwin's Embedding Theorem [S]. The connectivity condition on the map is "best possible," except that some work of Haefliger suggests that homological connectivity may be sufficient. Relaxing the connectivity condition on the map gives rise to an obstruction theory which will be described in a future paper. To what extent the connectivity condition on the manifold is really necessary I have no idea. The proof of the Embedding Theorem is a lengthy process done by shifting ƒ to general position and then applying a sequence of modifications each of which is fixed outside a PL ball [2] . 536