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Algebraic $K$–theory
over the infinite dihedral group: an algebraic approach
James F Davis, Qayum Khan and Andrew Ranicki |
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Algebraic & Geometric Topology 11 (2011) 2391–2436
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Abstract |
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Two types of Nil-groups arise in the codimension 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 –theory relating the two types of Nil-groups. The infinite dihedral group is a free product and has an index 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 –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. |
Keywords
Nil group, $K$–theory, Farrell–Jones Conjecture
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Mathematical Subject Classification 2000
Primary: 19D35
Secondary: 57R19
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References |
Publication
Received: 17 August 2010
Revised: 28 June 2011
Accepted: 26 July 2011
Published: 5 September 2011
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Authors |
| James F Davis | |
| Department of Mathematics Indiana University Bloomington IN 47405 USA |
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| Qayum Khan | |
| Department of Mathematics University of Notre Dame Notre Dame IN 46556 USA |
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| Andrew Ranicki | |
| School of Mathematics University of Edinburgh Edinburgh EH9 3JZ UK |
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