# Proof of the stable homeomorphism conjecture

I would like to get a demonstration of the stable homeomorphism conjecture (SHC$_n$), stating that any orientation-preserving homeomorphism $\mathbb{R}^n \rightarrow \mathbb{R}^n$ is stable in dimension $n$, for $n \neq 4$.

According to several sources, for example the upvoted answer to the post connectivity of the group of orientation-preserving homeomorphisms of the sphere , Robion Kirby proved it for $n > 4$ in the following paper http://www.maths.ed.ac.uk/~aar/papers/kirby.pdf .

("After that came R. Kirby (and L. Siebenmann) in 1968, who proved (using the results of surgery by Wall et al), that SHC$_n$ (hence AC$_n$) is true in all dimensions $n>4$.")

People often say in other papers that Kirby proves in fact in his paper not the SHC$_n$ for $n > 4$ but rather the annulus conjecture (AC$_n$) which would be equivalent to SHC$_n$, the equivalence being proved hypothetically in the paper of Brown and Gluck about "Stable structures on manifolds" (on JSTOR : https://www.jstor.org/stable/1970482?seq=1#fndtn-page_scan_tab_contents).

My two questions are the following : How can you justify precisely that :

1) Kirby proved AC$_n$ for $n > 4$ in his paper?

2) Brown and Gluck proved that AC$_n \Leftrightarrow$ SHC$_n$ for $n > 4$?

(I would like precise theorems in the papers and why they prove the results because after reading them a few times, I can't answer these two questions.)

PS : Nonetheless, if you have/know other research paths to prove SHC$_n$ for $n \neq 4$ don't hesitate, since it's my goal for all of this!

For your first question, yes Kirby did prove the n-dimensional annulus conjecture AC$_n$ for $n>4$ in the 1969 Annals paper that you cite. The only reason this might not be clear from reading the paper is that the original version of the paper proved AC$_n$ only assuming another statement that he calls HT$_n$ (namely the assertion that the PL structure on the $n$-torus is unique up to PL homeomorphism). But then Kirby added a note that appears in the published version of the paper, a note dated 1 December 1968, giving the argument that proves AC$_n$ for $n>4$ unconditionally. This argument uses a couple new results that had been proved by other people since the original version of the paper. Kirby's paper contains also a second added note dated 15 April 1969 which says that Siebenmann had just found an argument showing that HT$_n$ is in fact false for $n>4$. However, the first added note explains that the full strength of HT$_n$ is not needed, and that it suffices to assume the weaker version of HT$_n$ that asserts only that an arbitrary PL structure on the $n$-torus becomes standard after lifting it to a suitable finite-sheeted covering space. This weaker version had just been proved (for $n>4$) by Wall and Hsiang-Shaneson.
Actually Kirby does not deal directly with the annulus conjecture but rather with the stable homeomophism conjecture SHC$_n$. Thus the paper proves that the weaker version of HT$_n$ implies SHC$_n$ when $n>4$. Then Kirby quotes a 1964 result of Brown and Gluck that SHC$_n$ implies AC$_n$.
For your second question, Kirby states that Brown and Gluck also showed that AC$_k$ for all $k\leq n$ implies SHC$_n$. This of course is not the same as saying that AC$_n$ implies SHC$_n$ for a fixed $n$. However, Kirby's paper does not seem to use anything about AC implying SHC since he deals directly with SHC$_n$. This is fortunate since AC$_n$ for $n=4$ was only proved a number of years later by Quinn.
One other small comment: Kirby's note of 1 December 1968 contains a typographical error that might cause some confusion. He says that SHC$_n$ is a classical result for $n\neq 3$ when he must have meant $n\leq 3$.