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Algebraic and geometric splittings of the K- and L-groups of polynomial extensions. (English) Zbl 0585.57019

Math. Gottingensis, Schriftenr. Sonderforschungsbereichs Geom. Anal. 34, 45 p. (1985).
Novikov and the author obtained an algebraic proof of the splitting theorem \(L^ s_ n(\pi \times {\mathbb{Z}})=L^ s_ n(\pi)\oplus L^ h_{n-1}(\pi)\) for the Wall surgery obstruction groups, originally proved geometrically by Shaneson. In this work it is shown that the natural map \(L^ s_ n(\pi \times {\mathbb{Z}})\to L^ s_ n(\pi)\) of the geometric splitting is not quite the same as the algebraic splitting map induced by the natural projection of groups \(\pi\) \(\times {\mathbb{Z}}\to \pi\). Similar considerations apply to the Bass-Heller-Swan splitting of Wh(\(\pi\) \(\times {\mathbb{Z}})\).

MSC:

57R67 Surgery obstructions, Wall groups
18F25 Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects)