The homology cylinder group is the monoid of homology cylinders modulo homology cobordism. I wonder whether there is any known finite order element in this group.

$\begingroup$ Could you explain which dimension(s) you're interested in? Since there's a 3manifold tag, presumably the cylinders or their boundaries have dimension 3? $\endgroup$ – Ian Agol May 11 '12 at 14:18

$\begingroup$ @Agol: I think he is referring to 3dimensional cylinders with surface boundary. $\endgroup$ – Jim Conant May 11 '12 at 14:42

$\begingroup$ @Agol: yes, Conant is right. $\endgroup$ – hopflink May 11 '12 at 14:56
Certainly there is a lot of $2$torsion. One easy way to construct such an example is to look at the embedding of the string link cobordism group on $n$ strands into the homology cobordism group of genus $n$. (Levine explains this embedding in his paper "Homology cylinders: an enlargment of the mapping class group." See Theorem 4 of that paper.) The string link group has lots of $2$torsion. For example just tie an amphicheiral knot (which is not slice) into one of the strands. In fact the knot concordance group has infinitely many $\mathbb Z_2$ summands which inject into the homology cylinder group.

$\begingroup$ Thank you! You helps me a lot. Is there any other example, for surfaces with boundary ? $\endgroup$ – hopflink May 11 '12 at 14:16

1$\begingroup$ @HJ: this construction is for surfaces with one boundary component. I am sure there are other examples too. This paper arxiv.org/abs/0909.5580 shows there are infinitely many $\mathbb Z_2$ invariants, which should be realizable by actual homology cylinders, though I don't know off the top of my head. $\endgroup$ – Jim Conant May 11 '12 at 14:42

$\begingroup$ @Conant: Would there be more example for more boundary componant? $\endgroup$ – hopflink May 12 '12 at 7:18

$\begingroup$ @HJ: Sure. The map from string links to homology cobordisms extends to arbitrary numbers of components. Also, a string link is a homology cobordism of a planar surface so you get lots of direct examples that way. $\endgroup$ – Jim Conant May 12 '12 at 12:15