Certain (pre) simplicial sets have torsion-free homology? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T05:34:01Z http://mathoverflow.net/feeds/question/84395 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/84395/certain-pre-simplicial-sets-have-torsion-free-homology Certain (pre) simplicial sets have torsion-free homology? Shaun Ault 2011-12-27T17:08:50Z 2011-12-27T17:08:50Z <p>Suppose \$X\$ is a pre-simplicial set defined on a finite vertex set \${v_1, v_2, \ldots, v_k}\$ recursively as follows:</p> <ol> <li><p>Let \$X_1\$ have a single vertex \${v_1}\$.</p></li> <li><p>For \$i > 1\$, \$X_i\$ is obtained from \$X_{i-1}\$ by attaching the cone of a subcomplex \$S \subseteq X_{i-1}\$, \$X_i = X_{i-1} \cup CS\$, where we label the apex of \$CS\$: \$v_i\$.</p></li> <li><p>\$X = X_k\$.</p></li> </ol> <p>The point of the construction is that \$X\$ is built by successively including more simplices, and never identifying any of them. My question is this: Is \$H_*(X)\$ torsion-free? Small examples seem to indicate 'yes.' A cursory search of the internet does not turn up any results.</p> <p>Note, if a simplicial complex \$Y\$ is merely torsion-free, then \$Y \cup CS\$ need not be torsion-free, so the recursive construction of \$X\$ has to play a major role in preventing torsion (if the answer to the question turns out to be yes).</p> <p>Any references or ideas would be greatly appreciated!</p>