Are there lots of integer homology three-spheres? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T21:23:10Z http://mathoverflow.net/feeds/question/87393 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/87393/are-there-lots-of-integer-homology-three-spheres Are there lots of integer homology three-spheres? unknown (google) 2012-02-03T01:49:32Z 2012-02-03T07:33:26Z <p>The problem of counting combinatorial three-spheres with $N$ simplices has implications for some partition functions in physics (see a <a href="http://www.ams.org/mathscinet-getitem?mr=2810852" rel="nofollow">paper</a> 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 $C&lt;\infty$. According to the paper by Benedetti and Ziegler, providing such a bound is an open problem.</p> <p>One could ask whether a stronger property is true, namely whether the number of combinatorial <em>integer homology</em> three-spheres on $N$ simplices is bounded by $C^N$ for some $C&lt;\infty$. 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).</p>