TITLE: Small simplicial complexes with torsion in their homology?
QUESTION [35 upvotes]: Fix a prime $p$.  What is the smallest integer $n$ so that there is a simplicial complex on $n$ vertices with $p$-torsion in its homology?
For example, when $p=2$, there is a complex with 6 vertices (the minimal triangulation of the real projective plane) with 2-torsion in its homology.  I'm pretty sure that it's the smallest possible: with 5 or fewer vertices, there should be no torsion at all.  When $p=3$, there is a complex with 9 vertices (a triangulation of the mod 3 Moore space, for instance) with 3-torsion.  Is there one with 8 vertices?  With $p=5$, there is a complex with 11 vertices, found by randomly testing such complexes on my computer.
We can refine this: fix $p$ and also a positive integer $d$.  What's the smallest $n$ so that there is a simplicial complex $K$ on $n$ vertices with $p$-torsion in $H_d(K)$?  Or we can turn it around: for fixed $n$, what kinds of torsion can there be in a simplicial complex on $n$ vertices?
(A paper by Soulé ("Perfect forms and the Vandiver conjecture") quotes a result by Gabber which leads to a bound on the size of the torsion for a fixed number $n$ of vertices; however, this bound is far from optimal, at least for small $n$.)

REPLY [15 votes]: Andrew Newman just posted a preprint to the arXiv, showing that for every prime $p$ and $d \ge 2$, you can get $p$-torsion in homology $H_{d-1}(K)$ with only $O(\log^{1/d} p)$ vertices. (The implied constant depends on $d$, but not on $p$.) This is best possible, up to a constant factor.
His construction starts with something similar to what Speyer describes above, which gets you a complex with $O( \log p)$ vertices. Then he applies the probabilistic method, taking a certain carefully chosen random quotient of the complex, gluing together vertices in a random way. This doesn't affect torsion in homology. The problem is that it might not result in a simplicial complex. But Newman uses the Lovász local lemma to show that with positive probability, it does. Hence there exists a vertex identification that works.