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Let $M$ be a compact contractible manifold, $X\subset\partial M$ and $C_X$ the cone over $X$.

Question: Is it true that $C_X$ embeds in $M$ with its boundary $\partial C_X$ mapped to $X\subset \partial M$?

I am mostly interested in the piecewise linear case, that is, $M$ is a PL manifold, $X$ is a simplicial complex in $\partial M$, the embedding is a PL map, etc. I am also mostly interested in the case when $M$ is a 4-manifold, but a general answer is welcome too.

Note that the answer is "Yes" if $M$ is a ball.

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  • $\begingroup$ Don't non-slice knots in $S^3 = \partial D^4$ provide counterexamples? $\endgroup$
    – Neal
    Commented Sep 1, 2022 at 17:39
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    $\begingroup$ @Neal I am working in PL, where the embedding does not have to be locally flat. $\endgroup$
    – M. Winter
    Commented Sep 1, 2022 at 17:40

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Not in the PL case - this follows from the results of "Knot concordance in homology cobordisms" by Hom, Levine, and Lidman.

They prove that for many pairs of a 3-manifold $Y$ and knot $K \subset Y$, any contractible 4-manifold with boundary $Y$ does not contain a PL embedded disc with boundary $K$ even though $Y$ does in fact bound contractible 4-manifolds.

A particular example given in the paper is taking your $M$ to be a contractible 4-manifold with boundary $-1/2$ surgery on the right handed trefoil, and $X = K$ to be the core of the surgery torus. Then $X$ does not bound a PL disc in $M$.

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    $\begingroup$ Great source! One can't get clearer: "Resolving a conjecture of Zeeman [Zee64], Akbulut [Akb91] proved that the same need not hold for an arbitrary contractible 4-manifold: he exhibited a contractible 4-manifold $X$ and a knot $K\subset\partial X$ such that $K$ does not bound a PL disk (even with singularities) in $X$.", where the relevant reference is "Selman Akbulut, A solution to a conjecture of Zeeman". $\endgroup$
    – M. Winter
    Commented Sep 1, 2022 at 17:52
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    $\begingroup$ Thanks for your answer. Do you have some intuition of how this fails? Using a collar I can move the knot to the interior of $M$, and since $M$ is simply connected I can contract the knot to a point. Why does this fail to give me an embedded cone? Is it that the knot will need to pass through itself to contract, or is this not working in PL for some reason, ... ? $\endgroup$
    – M. Winter
    Commented Sep 1, 2022 at 21:03
  • $\begingroup$ And any chance that the following still works?: suppose $C_X$ embeds in $M$ as desired. Is there a neighborhood $N\subset \partial M$ of $X$ whose cone $C_N$ can be embedded into $M$ as well? $\endgroup$
    – M. Winter
    Commented Sep 1, 2022 at 21:44
  • $\begingroup$ Yes, the issue is that the knot might need to pass through itself to contract. I don't know the status of the problem if you leave the PL category, but it could be the case that there is always a topological disc as well. $\endgroup$
    – William Ballinger
    Commented Sep 2, 2022 at 0:24

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