6
$\begingroup$

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!

$\endgroup$

1 Answer 1

11
$\begingroup$

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 $n\leq 3$.

$\endgroup$
0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.