Surgery transfer. (English) Zbl 0677.57012

Lück, W.; Ranicki, A.

Algebraic topology and transformation groups, Proc. Conf., Göttingen/FRG 1987, Lect. Notes Math. 1361, 167-246 (1988).

From the introduction of the paper: Given a Hurewicz fibration \(F\to E\xrightarrow{p}B\) with fibre an \(n\)-dimensional Poincaré complex \(F\) we construct algebraic transfer maps in the Wall surgery obstruction groups \[ p^ *: L_m(\mathbb{Z}[\pi_1(B)])\to L_{m+n}(\mathbb{Z}[\pi_1(E)])\quad (m\geq 0) \] and prove that they agree with the geometrically defined transfer maps. The construction of the quadratic \(L\)-theory transfer maps is by a combination of the algebraic surgery theory of Ranicki and the method used by Lück to define the algebraic \(K\)-theory transfer maps for a fibration with finitely dominated fibre \(F\). The authors promise computations of the composition of \(p^*\) with \(p_*\) (the change of rings map) on either side, as well as some vanishing results in future work.
[For the entire collection see Zbl 0646.00011.]

Reviewer: Karl Heinz Dovermann (Honolulu)

MSC:

57R67	Surgery obstructions, Wall groups
55R12	Transfer for fiber spaces and bundles in algebraic topology
18F25	Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects)
55R10	Fiber bundles in algebraic topology

Keywords:

Hurewicz fibration with fibre an n-dimensional Poincaré complex; geometric transfer maps; algebraic transfer maps; Wall surgery obstruction groups; quadratic L-theory transfer maps; algebraic surgery; algebraic K-theory transfer maps

Citations:

Zbl 0646.00011