A topological disk in a 4-manifold can be approximated by piecewise linear disks
Gerard A. Venema · Bulletin of the American Mathematical Society · 1977
Several approximation theorems for embeddings of codimension 2 cells are announced here and the proofs are outlined.More detailed proofs will appear elsewhere [5]. Introduction.Our main theorem asserts that any topological embedding of a disk (2-cell) in a piecewise linear 4-manifold can be approximated arbitrarily closely by locally flat, piecewise linear embeddings.For codimension 2 cells in general, we do not prove as strong a theorem.If M" is a piecewise linear (PL) manifold and the topological embedding D: I n ~2 -• M n has the property that there is some open set UC I n ~~2 such that D\Ucan be e-approximated for every e > 0, then we show that D can be e-approximated for every e > 0. A corollary is that a piecewise linear, codimension 2 cell can be approximated by locally flat (n -2)-cells in all dimensions.If D: I n ~2 -• Af 1 is the topological embedding, the approximation can be chosen to agree with D on dl n ~~2 in both the theorems providing that D\dl n ~2 is PL and LHdI n ~2) C Int Af.This can be accomplished simply by pushing the boundary of the approximation to the boundary of D with a small ambient isotopy~ using [2] in case n = 4 and [3] in case n > 5.However, if Dtfl"" 2 ) C 3Af, the approximation cannot agree with D on the boundary.It is also not possible to replace e > 0 with a function e(x) > 0 with e(x) -• 0 as x -• 9Af.For example, the cone over the trefoil knot in the boundary of E\ cannot be approximated in this way.In fact, the trefoil knot does not bound any locally flat PL disk in E\ [4], 2. Statement of the theorems.THEOREM 1. IfD: I 2 -* M 4 is a topological embedding of a disk into a PL 4-manifold, then D can be e-approximated by a locally flat PL embedding E:I 2 -+M 4 for every e > 0. THEOREM 2. Suppose M 1 is a PL n-manifold and D: I n ~2 -• M n is a topological embedding.If there exists an open set U C l n ~~2 such that D\U has AMS (MOS) subject classifications (1970). Primary 57A15, 57A35, 57C55, 57C35; Secondary 57C30.