Collaring and $(n-1)$-manifold in an $n$-manifold
C. L. Seebeck · Transactions of the American Mathematical Society · 1970
C. L. SEEBECK UK1) 1. Introduction.If M is a locally flat two-sided PL m-manifold in a PL (m+1)manifold N then clearly [2] M can be approximated pointwise by locally flat embeddings from either side.Using a powerful result of Edwards and Kirby [7] we show conversely that M has a collar on one side if M can be approximated by locally flat embeddings from that side.As an application it follows that M is locally flat (even if M is one-sided in N) if N\M is 1-LC at each point of M, M can be approximated by locally flat embeddings, and m 5:4.Let / denote the interval [0, 1] and Id the identity mapping.Throughout we assume that M is a closed PL w-manifold, N is a PL n-manifold, n = m+\, and M is topologically embedded in A'0 with two sides.We choose a metric denoted by don Nand on Mx [ -1, 1 ] we choose the product metric p.In case A, Bare subsets of N and h is a homeomorphism of N we say that h is an e-push of (N, A) keeping B fixed if there is an isotopy ht of N such that h0 = ld, hx = h, and for each t e I ht is the identity on B and outside the ¿-neighborhood of A and d(ht, Id) 0 there is a 8>0 such that ifh : M x [-%, %] -> Mx [-1, 1] is an embedding within 8 o/Id|Mx [ -\, %] then there is an isotopy gt: Mx [-\, \] ->Mx [-1,1] such that g0 = h, gx\M x0 = h\Mx0, and for each tel gt\Mx {-i, i) = h\Mx{-i, i) and P(gt, Id) 0, and equal the identity on Mx [-1,0].Then, given e'>0, if« is small enough and h(M x0)<=^ M x[-1,0] then H=kfi1k~1\Mx [0, t0] is an embedding satisfying H\Mxt0 = Id\Mxt0, H\MxO = h\MxO, and diam H(xx[0,to])