In the same paper where [Milnor introduced](https://www.maths.ed.ac.uk/%7Ev1ranick/papers/micro001.pdf) 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?



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


[Rourke-Sanderson](https://link.springer.com/article/10.1007/BF01402954) 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](http://people.math.harvard.edu/%7Ekupers/teaching/272x/book.pdf). 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.


