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?



Recognizing the 4-sphere and the Adjan--Rabin theorem
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. 


