In Levine's paper, Homology cylinders: an enlargement of the mapping class group

Question

In the paper, Homology cylinders: an enlargement of the mapping class group, of J.Levine, I have a question in the proof of Theorem 4. The theorem says that there's injection $\hat\Phi$ from the concordance group $\mathcal{S}_g^\mathrm{fr}$ of framed $g $-string links to the homology cobordism group $\mathcal{H}_g$ of homology cylinders over $\Sigma_{g,1}$.

To prove this, he showed there is a map $\rho$ from a subset of $\mathcal{H}_g$ (containing the image of $\hat\Phi$) to $\mathcal{S}_g^{\mathrm{fr}}$, which gives $\rho \circ \hat\Phi = \mathrm{id}$.

But I cannot check the well-definedness of $\rho$, i.e., that if two are homology cobordant, then the images via $\rho$ are concordant.

Could anyone help me? Thank you.

Answer 1

In the definition of $\rho$ Levine used a homology cylinder $M$ to construct the complement $X$ of a string link in a homology 3-ball, by cutting out $\Sigma_{g,1} \times I$ from $D_g \times I$ and gluing in $M$. Given a homology cobordism from $M$ to $N$, attach the product cobordism of $X \setminus M$ to the homology cobordism. This should be a homology cobordism rel boundary of the exteriors of two string links, one produced from $M$ and one from $N$, in a homology $B^3 \times I$. Glue back in $\bigsqcup_m \nu(D^1 \times I)$, where $m$ is the number of components of the string link. The collection of the cores $D^1 \times I$ are then a concordance between two string links in a homology $B^3 \times I$. I hope.