On Chernavsky's theorem about the union of codimension 1 cells

The Schoenflies Theorem tells us that a codimension 1 sphere in $\mathbb R^n$ that is locally flat at all its points is topologically flat. After this was proved independently by Mazur and Brown, Cantrell improved the result for spheres of codimension 1 in $\mathbb R^n$ for $n>3$. That is: an embedded sphere has no isolated singularities where it fails to be locally flat. For $n=3$ there are famous counterexamples by Fox–Artin (1946). This theorem was generalised further by Chernavski (1966) for $n>4$ using engulfing, and independently by Kirby (1967). The generalisation may be stated as:

Let $q: B^{n-1}\to \mathbb R^n$ be an embedding that is flat on both semi-discs $B^{n-1}_+$ and $B^{n-1}_-$. Then $q$ is topologically flat.

We will discuss how this implies Cantrell's theorem and if time permits Chernavsky's 2006 paper on this result.

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