# Is there a torsion element in the homology cylinder group?

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.

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Could you explain which dimension(s) you're interested in? Since there's a 3-manifold tag, presumably the cylinders or their boundaries have dimension 3? –  Ian Agol May 11 '12 at 14:18
@Agol: I think he is referring to 3-dimensional cylinders with surface boundary. –  Jim Conant May 11 '12 at 14:42
@Agol: yes, Conant is right. –  hjjang 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.
@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. –  Jim Conant May 11 '12 at 14:42