Double \(L\)-groups and doubly slice knots (Q507052)

In this paper, the author introduces and studies the double \(L\)-groups, as a refinement of the torsion algebraic \(L\)-groups due to \textit{A. Ranicki} [Proc. Lond. Math. Soc. (3) 40, 87--192 (1980; Zbl 0471.57010); ibid. 40, 193--283 (1980; Zbl 0471.57011); Exact sequences in the algebraic theory of surgery. Princeton, New Jersey: Princeton University Press; University of Tokyo Press (1981; Zbl 0471.57012)]. Based on Ranicki's algebraic theory of surgery, which treats chain complexes equipped with some structure capturing algebraic symmetries, such as Poincaré duality, and a notion of an algebraic cobordism, the author introduces the notion of an algebraic double cobordism, and for a ring \(A\) with involution, a multiplicative subset \(S\) and a localization \(A \hookrightarrow S^{-1}A\), the author defines algebraic double cobordism groups of various types: the symmetric double \(L\)-groups \(DL^n(A)\), torsion symmetric double \(L\)-groups \(DL^n(A, S)\), and ultraquadratic double \(L\)-groups \(\widehat{DL}_n(A)\). More precisely, the double \(L\)-groups are equipped with a central unit \(\epsilon \in A\) with \(\epsilon \overline{\epsilon}=1\), where \(\overline{\epsilon}\) is the image of \(\epsilon\) by the involution, and denoted by \(DL^n(A, \epsilon)\) etc. The main results are as follows. For a ring \(A\) with involution with homological dimension \(0\), \(DL^{2k+1}(A, \epsilon)=0\) and \(DL^{2k}(A, \epsilon) \cong DL^0(A, (-1)^k\epsilon)\) for a non-negative integer \(k\), and a similar result holds true for the ultraquadratic case. For a ring \(A\) with involution, \(\widehat{DL}_0(A)\) is isomorphic to a certain group called the double Witt group of Seifert forms, and if there exists a central of \(A\) satisfying a certain condition, \(DL^0(A, S)\) is isomorphic to a certain group called the double Witt group of linking forms. As an application, by using a chain complex knot invariant called the Blanchfield complex, the author defines a homomorphism \(\sigma^{DL}\) from the \(n\)-dimensional double knot-cobordism group to the double \(L\)-group \(DL^{n+1}(\Lambda, P)\), where \(\Lambda=\mathbb{Z}[z, z^{-1}, (1-z)^{-1}]\) and \(P\) is the set of Alexander polynomials. An \(n\)-knot \(K: S^2 \hookrightarrow S^{n+2}\) is called doubly slice if it is equivalent to the form of the intersection of an \((n+1)\)-unknot and the equator \(S^{n+2}\subset S^{n+3}\), and \(K\) is called stably doubly slice if the connected sum \(K\#J\) is doubly slice for some doubly slice \(n\)-knot \(J\), and \(K\) is stably doubly slice if and only if \(K\) vanishes in the double knot-cobordism group. The author shows that for a stably doubly slice \(n\)-knot \(K\) (\(n \geq 1\)), the double \(L\)-class \(\sigma^{DL}(K) \in DL^{n+1}(\Lambda, P)\) of the Blanchfield complex vanishes, and similar results hold true for the Witt classes of the Blanchfield form, and the Witt class of any Seifert form.