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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:Ω 1 Ω 2 of open domains in n is quasiconformal if there is a K>0 such that for each x

lim ¯ r0 maxh ( x ) - h ( y ) ; |x-y| = r minh ( x ) - h ( y ) ; |x-y| = r:=K(x)<K·

Let M be a compact oriented quasiconformal manifold of even dimension 2l. Let γ be the /2-grading of L 2 (M, l T * ) associated to a measurable bounded conformal structure on M. Let U be a neighborhood of the diagonal in M×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=HγH+γ with kernel L(x,y). Then the measure σ=tr( 2q+1 L) is a U 2q -local Alexander Spanier cycle of dimension 2q;

3. The homology class of σ among U r -local cycles, r=2q(6l+2), is independent of the choice of H;

4. The homology class of σ is equal to λ 2q (L 2l-2q [M]), where L is the Hirzebruch-Thom L-class and λ 2q =2 2q+1 (2πi) -q q!/2q!.

Reviewer: W.Lück (Mainz)

MSC:
57N99Topological manifolds