Round L-theory. (English) Zbl 0638.18003

Hambleton, I.; Ranicki, A.; Taylor, L.

J. Pure Appl. Algebra 47, 131-154 (1987).

Wall’s algebraic L-groups, given by quadratic forms, were expressed as algebraic cobordism groups of Poincaré chain complexes by A. Ranicki [Proc. Lond. Math. Soc., III. Ser. 40, 87-192 (1980; Zbl 0471.57010)]. In this article the round L-groups, defined by quadratic forms with even rank, are identified with algebraic cobordism classes of round Poincaré complexes, i.e. Poincaré chain complexes with vanishing Euler characteristic. Products \(L\quad *_ r(A)\otimes L\quad p_*(B)\to L\quad h_*(A\otimes B)\) are defined which induce Shaneson’s splitting. Morita equivalence and \(L_*(A\times B)=L\quad r_*(A)\oplus L\quad r_*(B)\) hold. In particular Wederburn’s theorem \({\mathbb{Q}}\pi =\prod \Pi_{m(j)}(D_ j)\) implies \(L\quad r_*({\mathbb{Q}}\pi)=\prod L\quad r_*(D_ j).\) Product formulas for surgery obstructions and Rothenberg sequences are established. They are used to compute \(L\) \(r_*({\mathbb{Z}})\).

Reviewer: W.Lück

MSC:

18F25

Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects)

Keywords:

round surgery obstruction groups; round Poincaré chain complex

Citations:

Zbl 0471.57010

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