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In the same paper where Milnor introduced the concept of microbundles, he gave the following definition. $M$ has a microbundle neighborhood in $N$ if there is a neighborhood $U$ of $M$ in $N$ and a retraction $r: U \to M$ so that $(M, U, i, r)$ is a microbundle (where here $i$ is the inclusion map).

In a remark, Milnor mentions that he does not know if every locally flat submanifold $M$ of a topological manifold $N$ has a microbundle neighborhood. It has been many years since that foundational paper, so I imagine that result is now known. Does anybody know the state of this?

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Not all locally flat submanifolds have a normal microbundle, but they do stably.

Rourke-Sanderson prove that there is a PL embedding $S^{19} \times I \to S^{29}$ with no topological normal microbundle.

Milnor, in Microbundles 1, showed that every submanifold $M \subset N$ stabilizes to give $M \times 0 \subset N \times \Bbb R^q$ with a microbundle neighborhood for some large $q>0$.

Hirsch, in "On normal microbundles", gives relatively simple proofs of this fact and a few related facts. $q$ is quadratic in $\dim N$ in his result; I don't know whether or not it is known that the optimal $q(\dim M, \dim N)$ is quadratic in the inputs.

The first place I look for references to these sorts of foundational questions is Sander Kupers' notes on diffeomorphism groups. I found the Rourke-Sanderson reference at the beginning of Chapter 28. He cites Brown for the stability claim, but does not provide a reference.

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  • $\begingroup$ Additional reference with a stronger claim of no topological normal bundle: Millett, Proc. Amer. Math. Soc. 20 (1969), 580-584 ams.org/journals/proc/1969-020-02/S0002-9939-1969-0246306-3/…. $\endgroup$
    – Igor Belegradek
    Commented Dec 7, 2020 at 17:28
  • $\begingroup$ @IgorBelegradek Thanks as always, Igor. Do you happen to know if there is a quadratic lower bound for the optimal $q$? $\endgroup$
    – mme
    Commented Dec 7, 2020 at 17:33
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    $\begingroup$ I haven't thought about the topological case much. I think corollary 5.5 in [Rourke-Sanderson, "On topological neighborhoods"] gives a normal disk bundle when for codimension $q$ of a locally flat $l$-dimensional submanifold if $q>l$. I expect embeddings below metastable range (i.e. when $l\ge 2q-2$) generally don't have normal bundles. $\endgroup$
    – Igor Belegradek
    Commented Dec 7, 2020 at 17:56
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    $\begingroup$ Here is a tentative argument for nonexistence of normal microbundles. Suppose the codimension $q\ge 3$ and the embedding is PL, and the submanifold $L$ is closed. If $R$ is the regular neighborhood of $L$, then the inclusion $\partial R\to R$ is a spherical fibration. If $q$ is odd, such fibrations have rational characteristic class in $H^{2q-2}(L;\mathbb Q)$, and one can find embeddings with nonzero class if $H^{2q-2}(L;\mathbb Q)\neq 0$. $\endgroup$
    – Igor Belegradek
    Commented Dec 7, 2020 at 18:43
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    $\begingroup$ One reference for stable existence is Essay IV, Appendix A of Kirby-Siebenmann. $\endgroup$
    – skupers
    Commented Dec 7, 2020 at 21:56

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