Algebraic K–theory over the infinite dihedral group: an algebraic approach

Davis, James F¹ ; Khan, Qayum² ; Ranicki, Andrew³

¹Department of Mathematics, Indiana University, Bloomington IN 47405, USA
²Department of Mathematics, University of Notre Dame, Notre Dame IN 46556, USA
³School of Mathematics, University of Edinburgh, Edinburgh, EH9 3JZ, UK

Algebraic and Geometric Topology, Tome 11 (2011) no. 4, pp. 2391-2436

Résumé

Two types of Nil-groups arise in the codimension 1 splitting obstruction theory for homotopy equivalences of finite CW–complexes: the Farrell–Bass Nil-groups in the nonseparating case when the fundamental group is an HNN extension and the Waldhausen Nil-groups in the separating case when the fundamental group is an amalgamated free product. We obtain a general Nil-Nil theorem in algebraic K–theory relating the two types of Nil-groups.

The infinite dihedral group is a free product and has an index 2 subgroup which is an HNN extension, so both cases arise if the fundamental group surjects onto the infinite dihedral group. The Nil-Nil theorem implies that the two types of the reduced Nil˜–groups arising from such a fundamental group are isomorphic. There is also a topological application: in the finite-index case of an amalgamated free product, a homotopy equivalence of finite CW–complexes is semisplit along a separating subcomplex.

Détail

DOI : 10.2140/agt.2011.11.2391

Keywords: Nil group, $K$–theory, Farrell–Jones Conjecture

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