Divisible homology classes in the special linear group of a number field
Let \(F\) be a number field. \textit{G. Banaszak} has shown that the subgroup, \(D(i)\), of divisible elements of \(K_i (F)\) is a finite abelian group of number-theoretic interest [Ann. Math., II. Ser. 135, No. 2, 325-360 (1992; Zbl 0756.11037); Compos. Math. 86, No. 3, 281-305 (1993; Zbl 0778.11066)]. In this paper, the authors look at the analogous subgroup, \(\overline D (i)\), of divisible elements of \(H_i (\text{SL} (F), \mathbb{Z})\), building on previous work of Arlettaz on the structure of \(H_i (\text{SL} (F), \mathbb{Z})\). The first author [Comment. Math. Helv. 61, 556-564 (1986; Zbl 0621.20028)] has shown that \(H_i (\text{SL} (F); \mathbb{Z})\) is a direct sum of a free abelian group of finite rank and a torsion group. It follows that \(\overline D(i)\) is a torsion abelian group. In the first section of the present paper the authors show that \(\overline D(i)\) is in fact of finite exponent. In section 2 of the paper the authors consider the Hurewicz homomorphism \(h_i : K_i (F) \to H_i (\text{SL} (F); \mathbb{Z})\) and the induced map \(h_i : D(i) \to \overline D(i)\). For a prime number \(\ell\), let \(D(i)_\ell\) (resp. \(\overline D(i)_\ell) \) denote the \(\ell\)-torsion subgroup of \(D(i)\) (resp. \(\overline D(i))\). The authors show that, for any \(n \geq 1\), the map \(h_{2n} : D(2n)_\ell \to \overline D(2n)_\ell\) is a split injection if \(\ell > n \). (The localization exact sequence of \(K\)-theory shows that \(D(i) = 0\) for \(i\) odd.) They also show that \(\overline D (i)_\ell\) may contain nontrivial elements which do not belong to the image of \(h_i : K_i (F) \to H_i (\text{SL} (F); \mathbb{Z})\). In the final section of the paper they prove the following vanishing theorem: If \(N\) is a positive integer and \(\ell\) a prime number greater than \(N\) such that \(D(2n)_\ell = 0\) for \(1 \leq n \leq N\), then \(\overline D(i)_\ell = 0\) for \(1 \leq i \leq 2N\).
scientific article; zbMATH DE number 896557