Let $D\subset\Bbb R^4$ be a PL-embedded 2-dimensional disc. Let $N=D+K$ be a thickening of the disc, where $K$ is some sufficiently small 4-dimensional PL-ball and "$+$" means Minkowski addition.
Question: is $N$ homeomorphic to a 4-dimensional ball?
This could be non-trivial due to points inside the disc not being locally flat in the embedding.
I have an approach (see below), but this problem seems sufficiently natural that I expect this (or at least the tools) to be in some standard textbook. I am thankful for every reference or keyword (or short, self-contained, yet rigorous argument).
Some approach
Decompose $D$ into triangles $T_1,...,T_n$. We can assume that the triangles are ordered so that the intersection of $T_1\cup\cdots\cup T_i$ and $T_{i+1}$ is contractible. I then define a thickening $\tau_i:=T_i+K$ for each triangle and finalize by induction: $\tau_1$ is a 4-ball; and if $\tau_1\cup\dots\cup \tau_i$ is a 4-ball and the intersection between $\tau_1\cup\cdots\cup \tau_i$ and $\tau_{i+1}$ is sufficiently nice (which it is thanks to the ordering) then $\tau_1\cup\dots\cup \tau_{i-1}\cup \tau_{i+1}$ is a 4-ball as well.
This is still tedious to make rigorous (and I am not sure I thought of every detail) and I would rather cite a source, or would like to know whether this is considered folklore.
