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 Zap. Nauchn. Sem. LOMI, 1982, Volume 122, Pages 128–134 (Mi znsl4091)

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

V. G. Turaev

Abstract: 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.

Full text: PDF file (461 kB)

Bibliographic databases:
UDC: 515.162

Citation: V. G. Turaev, “$3$-dimensional homology spheres and links with Alexander polynomial 1.”, Investigations in topology. Part IV, Zap. Nauchn. Sem. LOMI, 122, "Nauka", Leningrad. Otdel., Leningrad, 1982, 128–134

Citation in format AMSBIB
\Bibitem{Tur82} \by V.~G.~Turaev \paper $3$-dimensional homology spheres and links with Alexander polynomial~1. \inbook Investigations in topology. Part~IV \serial Zap. Nauchn. Sem. LOMI \yr 1982 \vol 122 \pages 128--134 \publ "Nauka", Leningrad. Otdel. \publaddr Leningrad \mathnet{http://mi.mathnet.ru/znsl4091} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=661471} \zmath{https://zbmath.org/?q=an:0504.57001}