Satellite operators as group actions on knot concordance
A knot in the solid torus, called a pattern, induces a map on knots via the satellite construction. The corresponding satellite operators induced on concordance classes of knots have received broad attention. The article under review introduces a group action of strong homology cylinders (up to strong homology cobordism) on concordance classes of knots in homology spheres. The main result states that the satellite operators of strong winding number \(\pm1\) patterns are a restriction of this group action.NEWLINENEWLINEThere is a version of the result in the exotic, the topological and the \(R\)-homology category, where \(R\) is any localization of \(\mathbb{Z}\). The proof uses surgery techniques and some basic algebraic topology. Freedman's proof of the 4-dimensional topological Poincaré conjecture is used to inject the concordance classes of knots into the concordance classes of knots in homology spheres.NEWLINENEWLINEThe article under review nicely presents how a better understanding of satellite operators can be obtained from the group action described in the main theorem. For example, injectivity of the satellite operators of strong winding number \(\pm1\) patterns is deduced, a result which has previously been obtained by \textit{T. D. Cochran} and the authors [J. Topol. 7, No. 4, 948--964 (2014; Zbl 1312.57006)]. Furthermore, a criterion for the satellite operator of a strong winding number \(\pm1\) pattern to be surjective is given. In combination, infinitely many patterns (different from connected-sum patterns) with bijective satellite operators are constructed.NEWLINENEWLINEIn this context, it is important to mention a conjecture of Akbulut asserting that there exists a winding number \(\pm1\) pattern such that none of the corresponding satellite knots is slice (Problem 1.45 on [\textit{R. Kirby}, Proc. Symp. Pure Math., Vol. 32, Part 2, 273--312 (1978; Zbl 0394.57002)]). Such an example has recently been provided by Levine in the preprint \textit{Non-surjective satellite operators and piecewise-linear concordance} for the following categories: smooth, exotic, \(\mathbb{Z}\)-homology and \(\mathbb{Q}\)-homology.
scientific article; zbMATH DE number 6577071