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As far as I understand it follows from the work of M. Freedman that there exist locally flat embeddings of two dimensional surfaces in $\mathbb R^4$ that can not be smoothed in the class of locally flat embedding. (For definition of locally flat see http://en.wikipedia.org/wiki/Local_flatness)

I am curious if one can draw a realistic picture of such a surface. At least is it possible to draw an intersection of such a surface with a (linear) hyperplane in $\mathbb R^4$? Is the Hausdorff dimension of such an intersection equals $1$?

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Not quite what you want, but if you take a knot in $R^3$ which is topologically slice but not slice, e.g. the whitehead double of the trefoil, it bounds a locally flat continuously embedded disk in upper half-$R^4$, but does not bound a smooth such disk. So the intersection of this weird disk with a linear hyperplane is a smooth knot. – Paul Nov 28 at 15:54
Paul I understand that some of intersections of the surface with linear subspaces can be smooth, as in you example. I wonder if one can tell what would be the worst possible intersection for your example. How would it look like? – aglearner Nov 29 at 15:49
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 Coming at this with a lot of ignorance, this raises the following related question: are there locally-flat embeddings of surfaces in $\mathbb R^4$ that are not smoothable embeddings but such that the intersection with every hyperplane parallel to a given one are all sufficiently "nice" (either smoothly embedded curves, smoothly immersed curves, or finite sets of points (or unions of such things))? – Greg Friedman Dec 5 at 6:12

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