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



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> **Question:** Is it true that $C\_X$ embeds in $M$ with its boundary $\partial C\_X$ mapped to $X\subset \partial M$?
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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.



If $M$ is contractible manifold and $X\subset \partial M$, does the cone over $X$ embed in $M$?
Not in the PL case - this follows from the results of ["Knot concordance in homology cobordisms"](https://arxiv.org/abs/1801.07770) 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 $M$.


