Re: PL vs. PL locally flat

An isolated cone point doesn't have a smooth local product structure,
but it does have a piecewise-linear local product structure as long as
the link of the point is an unknotted codimension-2 piecewise-linear
sphere. For a point to be "locally a product" means there's a
neighborhood which is equivalent to the standard product, but the
meaning of "equivalent" depends on what category you're working in.
(That's what "charitable enough in your definition" means, I think.)
Every embedding of the 1-sphere in 2-space is locally flat in the
topological category, no matter how bad it looks (think of the
snowflake curve, for example), whereas no embedding of the 1-sphere in
2-space is nice enough to be locally flat in the smooth and
piecewise-linear categories simultaneously, although the circle and the
square do well enough for one category at a time. It's fairly easy to
triangulate a neighborhood of the corner of a square, and then change
the sizes of the triangles to straighten out the corner, and the same
can be done in the neighborhood of a locally flat piecewise-linear cone
point in 3 or 4 dimensions.

To construct a PL locally flat embedding (trivial or non-trivial) of
the 2-sphere or torus in 4-space, take your favorite smooth picture
(such as a movie of links in 3-space) and "square the circle": break
the embedding into flat pieces, and keep them connected along
reasonable corners.

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