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I am given a PL 2-disc $D\subset\Bbb R^4$ (everything PL from here on) and I need a "neighborhood" $N\simeq B^4$ (PL-homeomorphic to a 4-ball) so that $\partial N\cap D=\partial D$. If I got this right, then since $D$ is collapsible a regular neighborhood [1] will be a 4-ball; but unfortunately it contains $\partial D$ in its interior.

Question: Can one tweak the theory of regular neighborhoods to give the desired neighborhood?

Another approach: it should be possible to "push the not locally flat parts" of $D$ to the boundary, so there should be a tubular neighborhood $T$ of $\mathrm{int}(D)$ [2]. Does this work in PL? Do we have $T\simeq\mathrm{int}(D)\times B^2$? If so, I believe I can make it taper off towards the boundary and take the closure.

  • [1] Rourke, Sanderson, "Introduction to Piecewise-Linear Topology", Chapter 3
  • [2] Brown, "Locally flat imbeddings of topological manifolds"
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Pick a triangulation of $\mathbb R^4$ which contains $D$ as a subcomplex, make two barycentric subdivisions, and then pick the closure of all the simplexes that intersect the interior of $D$. This could be a promising candidate.

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  • $\begingroup$ While I agree that this is a promising candidate, I can't think of any standard tool that tells me that this is a 4-ball. E.g. let $D'$ be the subcomplex of $D$ obtained by deleting all triangles at the boundary, and assume that this is still a 2-disc. Is the neighborhood you describe a regular neighborhood of $D'$? I am not sure. $D$ could be weird at the boundary. $\endgroup$
    – M. Winter
    Commented Sep 11, 2022 at 16:37

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