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

作者：Davis James F^*; Khan Qayum; Ranicki Andrew

来源：Algebraic and Geometric Topology, 2011, 11(4): 2391-2436.

DOI：10.2140/agt.2011.11.2391

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) over tilde -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.

• 出版日期2011