Abstract
The computation of the knot cobordism groups reduces to the computation of the L-theory of various fields and integral domains. The main properties of the L-groups of a field with involution F are summarized in 37A. The computations of the L-groups of ℤ, ℤ m , \({\widehat {\Bbb Z}_m}\), ℚ and the localization exact sequence are used in 37B to prove a general result: the natural maps L* (A) → L* (ℚ ⊗ℤ A) are isomorphisms modulo 8-torsion for any additively torsion-free ring involution A (e.g. a group ring A = ℤ[π]). In Chap. 42 the version of this result for torsion L-groups will be used to prove that the natural maps
are isomorphisms modulo 8-torsion, for an appropriately defined U ℤ.
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© 1998 Springer-Verlag Berlin Heidelberg
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Ranicki, A. (1998). L-theory of fields and rational localization. In: High-dimensional Knot Theory. Springer Monographs in Mathematics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-12011-8_37
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DOI: https://doi.org/10.1007/978-3-662-12011-8_37
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-08329-7
Online ISBN: 978-3-662-12011-8
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