Quasiconformal mappings, operators on Hilbert space, and local formulae for characteristic classes. (English) Zbl 0840.57013

The paper deals in particular with the question whether one can construct a representative for the Hirzebruch-Thom-\(L\)-class on a quasiconformal manifold. Classically this can be done for a smooth Riemannian manifold, here only a quasiconformal structure shall be used. A quasiconformal manifold is a topological manifold with an atlas whose changes of coordinates are all quasiconformal homeomorphisms. A homeomorphism \(h : \Omega_1 \to \Omega_2\) of open domains in \(\mathbb{R}^n\) is quasiconformal if there is a \(K > 0\) such that for each \(x\) \[ \varlimsup_{r \to 0} {\max \biggl\{ \bigl |h(x) - h(y) \bigr |; |x - y |= r \biggr\} \over \min \biggl\{ \bigl |h(x) - h(y) \bigr |; |x - y |= r \biggr\}} : = K(x) < K. \] Let \(M\) be a compact oriented quasiconformal manifold of even dimension \(2l\). Let \( \gamma \) be the \(\mathbb{Z}/2\)-grading of \(L^2 (M, \wedge^l T^*_\mathbb{C})\) associated to a measurable bounded conformal structure on \(M\). Let \(U\) be a neighborhood of the diagonal in \(M \times M\). Then the main result of the paper says:
1. There is a locally constructed \(U\)-local Hodge decomposition \(H\);
2. Let \(H\) be a \(U\)-local Hodge decomposition and \(L = H \gamma H + \gamma\) with kernel \(L(x,y)\). Then the measure \(\sigma = \text{tr} (\wedge^{ 2q + 1} L)\) is a \(U^{2q}\)-local Alexander Spanier cycle of dimension \(2q\);
3. The homology class of \(\sigma\) among \(U^r\)-local cycles, \(r = 2q (6l + 2)\), is independent of the choice of \(H \);
4. The homology class of \(\sigma\) is equal to \(\lambda_{2q} (L_{2l - 2q} \cap [M])\), where \(L\) is the Hirzebruch-Thom \(L\)-class and \(\lambda_{2q} = 2^{2q + 1} (2 \pi i)^{-q} q!/2q!\).
Reviewer: W.Lück (Mainz)


57N99 Topological manifolds
Full Text: DOI Link