Approximating maps of codimension one spheres by embeddings.

The problem about approximating topological embeddings of polyhedra into PL manifolds by PL embeddings has a long and rich history. This will be a survey talk on embeddings of codimension one spheres into $S^3$. In particular, we shall discuss the status of two conjectures - the first one was made by Bing in 1950's and the second one was proposed in 1980's by the speaker: Every continuous map $f:S^2
ightarrow S^3$ with a 0-dimensional singular set can be approximated by embeddings. We shall describe results concerning both conjectures. Most recently, Brahm proved the special case of the second conjecture for the special case when the closure of the singular set of $f$ is 0-dimensional.
However, in general, the 0-dimensional singular set can be dense in $S^2$. We shall actually describe very pathological examples of such maps whose images can be 3-dimensional. We shall also discuss the status of these conjectures and present some further examples and related problems. In particular, we will see how this problem is connected to the problem of finding appropriate general positions for recognizing topological 3-manifolds which are analogues of the well-known Edwards' higher dimensional Disjoint Disks Property.