Correction to the article ``An infinite-rank summand of topologically slice knots'' (Q2331026): Difference between revisions

Let \(\mathcal{C}\) denote the smooth concordance group and let \(\mathcal{C}_{\mathrm{TS}}\) be the subgroup of \(\mathcal{C}\) consisting of topologically slice knots. The main result of the paper under review is the existence of a \(\mathbb{Z}^{\infty}\) summand in \(\mathcal{C}_{\mathrm{TS}}\). The existence of a \(\mathbb{Z}^{\infty}\) \textit{subgroup} in \(\mathcal{C}_{\mathrm{TS}}\) was known before: see [\textit{H. Endo}, Topology Appl. 63, No. 3, 257--262 (1995; Zbl 0845.57006)] and [\textit{J. Hom}, Comment. Math. Helv. 89, No. 3, 537--570 (2014; Zbl 1312.57008)]. It is instructive to compare this with \(\mathbb{Q}\) -- it has a \(\mathbb{Z}^{\infty}\) subgroup, but not a summand. \textit{M. Hedden}, \textit{S.-G. Kim} and \textit{C. Livingston} [``Topologically slice knots of smooth concordance order two'' (2012), \url{arXiv:1212.6628}] have established the existence of a \(\mathbb{Z}_2^{\infty}\) subgroup in \(\mathcal{C}_{\mathrm{TS}}\). Given a null-homologous knot \(K\) in a three-manifold \(Y\), \textit{P. Ozsváth} and \textit{S. Szabó} [Adv. Math. 186, No. 1, 58--116 (2004; Zbl 1062.57019)] and independently \textit{J. Rasmussen} [Floer homology and knot complements. PhD thesis. Harvard University (2003)] define a doubly-filtered chain complex \(CFK^{\infty}(Y, K)\) whose doubly-filtered chain homotopy type is an invariant of \(K\). When \(Y = S^3\), this complex is denoted by \(CFK^{\infty}(K)\). For knots in \(S^3\), using \(CFK^{\infty}\) one may define a smooth concordance invariant \(\tau\) [\textit{P. Ozsváth} and \textit{S. Szabó}, Geom. Topol. 7, 615--639 (2003; Zbl 1037.57027)] which gives a surjective homomorphism from \(\mathcal{C}\) to \(\mathbb{Z}\). \textit{C. Livingston} [Geom. Topol. 8, 735--742 (2004; Zbl 1067.57008)] has found topologically slice knots with \(\tau\)-invariant \(1\) thus establishing the existence of a \(\mathbb{Z}\) summand in \(\mathcal{C}_{\mathrm{TS}}\). This result was improved in [\textit{C. Manolescu} and \textit{B. Owens}, Int. Math. Res. Not. 2007, No. 20, Article ID rnm077, 21 p. (2007; Zbl 1132.57013)] to show that \(\mathcal{C}_{\mathrm{TS}}\) contains a \(\mathbb{Z}^2\) summand and in [\textit{C. Livingston}, Proc. Am. Math. Soc. 136, No. 1, 347--349 (2008; Zbl 1137.57015)] to show that it contains a \(\mathbb{Z}^3\) summand. In [\textit{J. Hom}, J. Topol. 7, No. 2, 287--326 (2014; Zbl 1368.57002)] the author uses \(CFK^{\infty}\) to define a smooth concordance invariant \(\varepsilon\) that takes values in \(\{ -1, 0, 1\}\). The article under review builds upon and extends results of [\textit{J. Hom}, Comment. Math. Helv. 89, No. 3, 537--570 (2014; Zbl 1312.57008)] by the author. In that paper the author defines \[ \mathcal{CFK} = \{CFK^{\infty}(K) | K \subset S^3\}/ \sim, \] where \(CFK^{\infty}(K) \sim CFK^{\infty}(K')\) if \(\varepsilon(K\# -K') = 0\). \(\mathcal{CFK}\) is an abelian group with the operation given by taking tensor products and the inverse given by taking a dual complex. Moreover, this group is totally ordered by \([CFK^{\infty}(K)]>[CFK^{\infty}(K')]\) iff \(\varepsilon(K\# -K') = 1\). In fact, a ``much bigger'' relation \(\gg\) can be defined and sets of knots whose complexes satisfy it are linearly independent in \(\mathcal{C}\). Infinitely many topologically slice knots \(\{K_i\}_{i=0}^{\infty}\) that satisfy \([CFK^{\infty}(K_i)] \gg [CFK^{\infty}(K_{i-1})]>0\) are found and thus the existence of a \(\mathbb{Z}^{\infty}\) subgroup in \(\mathcal{C}_{\mathrm{TS}}\) is established. In the paper under review the author also defines ``property A'' for elements in totally ordered groups and shows that a set of elements \(\{g_i\}_{i=0}^{\infty}\) in a group \(G\) that satisfy \(g_i \gg g_{i-1}>0\) and have property A give a set of linearly independent surjective homeomorphisms from \(G\) to \(Z\) and thus establish the existence of a \(\mathbb{Z}^{\infty}\) summand in \(G\). This is proven using the Hahn embedding theorem. After defining some new invariants \(a_i\) (that measure the length of certain arrows in the \(CFK^{\infty}\) complex) of knots with \(\varepsilon = 1\), a consideration of cases establishes that the knots \(\{K_i\}_{i=0}^{\infty}\) have property A and thus the main result is proven. Note that the kernel of Levine's homomorphism (see \textit{J. Levine} [Invent. Math. 8, 98--110 (1969; Zbl 0179.52401); addendum ibid. 8, 355 (1969); Comment. Math. Helv. 44, 229--244 (1969; Zbl 0176.22101)]) contains \(\mathcal{C}_{\mathrm{TS}}\) thus the main result also implies the existence of a \(\mathbb{Z}^{\infty}\) summand in the kernel of Levine's homomorphism.