If I understand my PL topology correctly (and please correct me if I don't), if $K$ is a $k-$complex and $n\ge 2k+2$, then any two PL embeddings $a,b\colon K\to \mathbb{R}^n$ are isotopic. Therefore, the regular neighborhoods of $a(K)$ and $b(K)$ are homeomorphic, and we can think about them as "the" regular neighborhood of $K$. 


Is it possible to describe this regular neighborhood without reference to a particular embedding? For instance, if $K$ is a manifold, then the regular neighborhood is homeomorphic to a product $K\times \mathbb{R}^{n-k}$, and I'd like something similar for complexes that aren't manifolds.



Can one construct a regular neighborhood without an ambient space?
One might think of the abstract regular neighbourhood as a canonical way of thickening $K$ to a handle decomposition of sufficiently high dimension. 


If $\dim K = 1$, then $K$ has a canonical $n$-dimensional thickening $M$ as soon as $n\ge 3$, where vertices and edges are replaced by 0-handles and 1-handles. Of course $M$ is a handlebody. Note that $M$ is canonical because we implicitly require $M$ to be orientable, that is $w\_1(M)=0$.


If $\dim K = 2$, then analogously $K$ has a canonical 5-dimensional thickening $W$ that has $w\_1(W) = w\_2(W)=0$. This can be proved by hand with standard low-dimensional topology techniques, see for instance Lemma 3.3 in this [paper](http://ms.mcmaster.ca/ian/papers/hkt_revision.pdf) of Hambleton, Kreck, and Teichner. Vertices, edges, and faces are thickened to 0-, 1-, and 2-handles.


As they say in this paper, the general case might be covered by this 1966 [paper](https://www.sciencedirect.com/science/article/pii/004093836690005X) of Wall, but I must admit I haven't read it.


