The problem of counting combinatorial three-spheres with $N$ simplices has implications for some partition functions in physics (see a paper by Benedetti and Ziegler for more background and references). To add some context, let's first make the observation that there are roughly order $C^{N\log N}$ combinatorial three-manifolds on $N$ simplices. Now there is a partition function in quantum gravity whose convergence depends on knowing that the number of combinatorial three-spheres on $N$-simplices is bounded by $C^N$ for some $C<\infty$. According to the paper by Benedetti and Ziegler, providing such a bound is an open problem.

One could ask whether a stronger property is true, namely whether the number of combinatorial *integer homology* three-spheres on $N$ simplices is bounded by $C^N$ for some $C<\infty$. Is this strengthened conjecture known to be false? (certainly it can't be known to be true, since the weaker statement about bounding the number of genuine three-spheres is still open).