$3$-dimensional homology spheres and links with Alexander polynomial~1.

V. G. Turaev

Zapiski Nauchnykh Seminarov POMI, Investigations in topology. Part IV, Tome 122 (1982), pp. 128-134

Résumé

The paper establishes a relationship between two key problems of $3$-dimensional topology: (a) the sliceness problem for knots and links with Alexander polynomial $\Delta=1$ and (b) the triviality problem for the kernel of the Rochlin homomorphism $R$ from the group of $\mathbb Z$-homologically cobordant, $\mathbb Z$-homology $3$-spheres onto $\mathbb Z/2$. The following statements are proved: If $\operatorname{Ker}\ne0$ then there exist a homology $4$-ball $V$ and a knot $k$ of genus $1$ in with $\Delta=1$ such that $k$ does not bound a locally flat disc in any homology $4$-ball $V'$ with $\partial V'=\partial V$. If $\operatorname{Ker} R\ne0$ then there exists a boundary link in $S^3$ with $\Delta=1$ which does not bound a set of nonintersected locally flat discs in any homology ball bordered by $S^3$. These results suggest that obstructions to sliceness of knots and links may lie in $\operatorname{Ker} R$. On the other hand it is possible that $\operatorname{Ker} R=0$ and the present results might be used in a proof of this equality.