CONSTRUCTING NEAR-EMBEDDINGS OF CODIMENSIONONE MANIFOLDS WITH COUNTABLE DENSE SINGULARSETS

摘要

We present, for all n 3, very simple examples of con-tinuous maps f : M~(n-1) M~nfrom closed (n — 1)-manifolds M~(n-1)intoclosed n-manifolds M~nsuch that even though the singular setS(f)offiscountable and dense, the map f can nevertheless be approximated by anembedding, i.e. f is a near-embedding. In dimension 3 one can get even apiecewise-linear approximation by an embedding.