Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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.

share|improve this question
    
And I don't know why "$\mathcal{H}_g$ of homology cylinders over $\Sigma_{g,1}$" is shown as above -_-; –  MKS Jun 17 '13 at 8:21
add comment

1 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.

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.