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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.

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  • $\begingroup$ The global embedding can be constructed by a general position argument. Since you do not want to use it, you may describe a neighborhood, by embeddings of the stars at each vertex such that they agree on the common faces. Note that by the same theorem these star embeddings are all the same; in particular you may think that the embeddings of stars come from a global embedding and in particular they agree. $\endgroup$ – Anton Petrunin Jun 19 '18 at 19:57
  • $\begingroup$ I'll have to think about that. I think this probably works if the embeddings of the stars are based on a submersion, but I'm worried that there might be global issues -- for instance, if $K$ is a manifold, then this constructs a disc bundle over $K$, which might not be trivial. $\endgroup$ – Robert Young Jun 20 '18 at 17:25
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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 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 of Wall, but I must admit I haven't read it.

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  • $\begingroup$ Interesting! Thanks for the references. I think this also explains how the regular neighborhood depends on the ambient space -- if you embed into a different high-dimensional manifold, you get a thickening with Stiefel-Whitney classes coming from the ambient space. $\endgroup$ – Robert Young Jun 20 '18 at 17:42
  • $\begingroup$ @BrunoMartelli How about 4-dimensional thickening of 2-complex ? I have found paper of Krushkal, Embedding obstructions and 4-dimensional thickenings of 2-complexes, Proc of AMS, Volume 128, Number 12, 2000, ams.org/journals/proc/2000-128-12/S0002-9939-00-05458-7/… but I have not read it yet. In chapter 4 they construct 2-complex which cannot be embedded to R^4. $\endgroup$ – Marek Mitros Aug 24 '18 at 9:19

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