Celebratio Mathematica — Ranicki

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Works connected to Qayum Khan

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J. F. Davis, Q. Khan, and A. Ranicki: “Algebraic \( K \)-theory over the infinite dihedral group: An algebraic approach,” Algebr. Geom. Topol. 11 : 4 (2011), pp. 2391–2436. MR 2835234 Zbl 1236.19002 ArXiv 0803.1639 article

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 \( \widetilde{\mathrm{Nil}} \)-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.

James Frederic Davis	Related
Qayum Khan	Related