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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 $S^n$.

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

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.

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.

Question: Is the recognition problem for $S^4$ solvable?

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.

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?

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up vote 7 down vote accepted

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.

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.

Another closely-related problem would be an algorithmic Schoenflies theorem, to determine if a normal 3-sphere bounds a ball.

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A presentation with the same number of generators and relations is called balanced. 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.

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Thank you for the answer, I'll look into that. And thank you in particular for the remark on perfect groups! – Malte Mar 23 '12 at 14:32

Recognition of $S^4$ is listed as an open problem in the survey of Shmuel Winberger "Homology Manifolds" (page 1088): 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.

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The Hausmann--Weinberger example is good news. One might not have to solve the triviality problem for balanced groups then to get the S.P. Novikov theorem for $n=4$. – Malte Mar 23 '12 at 14:33

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