Recognizing the 4-sphere and the Adjan--Rabin theorem - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T08:34:17Z http://mathoverflow.net/feeds/question/91994 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/91994/recognizing-the-4-sphere-and-the-adjan-rabin-theorem Recognizing the 4-sphere and the Adjan--Rabin theorem Malte 2012-03-23T10:49:20Z 2012-03-23T19:33:32Z <p>The problem of recognizing the standard $S^n$ is the following: Given some simplicial complex $M$ with rational vertices representing a closed manifold, can one decide (in finite time) if $M$ is homeomorphic to $S^n$. </p> <p>For $n=1$, this is obvious, and for $n=2$, one can solve it by computing $\chi(M)$. A solution for $n=3$ is due to </p> <p>J.H. Rubinstein. An algorithm to recognize the 3-sphere. In Pro- ceedings of the International Congress of Mathematicians, vol- ume 1, 2, pages pp. 601–611, Basel, 1995. Birkhäuser.</p> <p>By a theorem of S.P. Novikov, the problem is unsolvable if $n\geq 5$. The idea is the following: By the Adjan--Rabin theorem, there is a sequence of super-perfect groups $\pi_i$ for which the triviality problem is unsolvable. Now construct homology spheres $\Sigma_i$ with fundamental groups $\pi_i$. If one can decide which of the $\Sigma_i$ are standard spheres, then one can solve the triviality problem for the fundamental groups.</p> <p>Question: Is the recognition problem for $S^4$ solvable?</p> <p>The problem with this proof of S.P. Novikov's theorem is that there is no result that asserts that for any given super-perfect group $\pi$ there is a homology $4$-sphere satisfying $\pi_1(\Sigma) = \pi$. However, Kervaire has proved that every perfect group with the same amount of generators and relators may be realized as the fundamental group of a homology $4$-sphere.</p> <p>Thus the question: Is there an improved Adjan--Rabin theorem that asserts the existence of a sequence of perfect groups $\pi_i$ with the same amount of generators and relators, the triviality problem of which is unsolvable?</p> http://mathoverflow.net/questions/91994/recognizing-the-4-sphere-and-the-adjan-rabin-theorem/91997#91997 Answer by HW for Recognizing the 4-sphere and the Adjan--Rabin theorem HW 2012-03-23T11:19:57Z 2012-03-23T11:19:57Z <p>A presentation with the same number of generators and relations is called <em>balanced</em>. The triviality problem for balanced presentations (indeed, the word problem for balanced presentations) is a major unsolved problem. Googling the phrase 'triviality problem for balanced presentations' will give lots of references. Note that you may automatically assume that your groups $\pi_i$ are perfect, since the class of perfect groups is recursive.</p> http://mathoverflow.net/questions/91994/recognizing-the-4-sphere-and-the-adjan-rabin-theorem/92005#92005 Answer by Misha for Recognizing the 4-sphere and the Adjan--Rabin theorem Misha 2012-03-23T12:05:16Z 2012-03-23T12:05:16Z <p>Recognition of $S^4$ is listed as an open problem in the survey of Shmuel Winberger "Homology Manifolds" (page 1088): <a href="http://www.maths.ed.ac.uk/~aar/homology/shmuel2.pdf" rel="nofollow">http://www.maths.ed.ac.uk/~aar/homology/shmuel2.pdf</a> with exactly the same reasoning that HW explained. Note that fundamental groups of homology 4-spheres need not be balanced (an example of Hausmann and Weinberger from 1984), still, nobody so far was able to exploit this. </p> http://mathoverflow.net/questions/91994/recognizing-the-4-sphere-and-the-adjan-rabin-theorem/92034#92034 Answer by Ryan Budney for Recognizing the 4-sphere and the Adjan--Rabin theorem Ryan Budney 2012-03-23T19:33:32Z 2012-03-23T19:33:32Z <p>As mentioned algorithmic 4-sphere recognition is an open problem. Since Rubinstein's solution to the 3-sphere recognition problem is so simple and elegant, perhaps the first thing you might guess is, why not try those techniques in dimension 4? Normal surfaces, crushing normal 3-spheres, searching for almost-normal 3-spheres. </p> <p>That theory is still in its infancy. Rubinstein and his former student Bell Foozwell have been developing normal co-dimension one manifold theory in triangulated manifolds. They have a "normalization" process that follows Rubinstein's general normal/almost-normal schema but it appears to do a fair bit of damage to the manifolds, so it's not clear to me if anything like this could eventually be used for 4-sphere recognition, but maybe some creative variant of the idea will work-out. </p> <p>Another closely-related problem would be an algorithmic Schoenflies theorem, to determine if a normal 3-sphere bounds a ball. </p>