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Zbl 0840.57013
Connes, Alain; Sullivan, Dennis; Teleman, Nicolas
Quasiconformal mappings, operators on Hilbert space, and local formulae for characteristic classes. (English)
[J] Topology 33, No.4, 663-681 (1994). ISSN 0040-9383

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 $\bbfR^n$ 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 $\bbfZ/2$-grading of $L^2 (M, \wedge^l T^*_\bbfC)$ 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:\par 1. There is a locally constructed $U$-local Hodge decomposition $H$;\par 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 $2q$;\par 3. The homology class of $\sigma$ among $U^r$-local cycles, $r = 2q (6l + 2)$, is independent of the choice of $H $;\par 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 $\lambda_{2q} = 2^{2q + 1} (2 \pi i)^{-q} q!/2q!$.
[W.Lück (Mainz)]

MSC 2000:: *57N99 Topological manifolds

Keywords: Hirzebruch-Thom-$L$-class; quasiconformal manifold; Hodge decomposition

Cited in: Zbl 1008.58020 Zbl 0840.57016

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