The paper deals in particular with the question whether one can construct a representative for the Hirzebruch-Thom--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 of open domains in is quasiconformal if there is a such that for each
Let be a compact oriented quasiconformal manifold of even dimension . Let be the -grading of associated to a measurable bounded conformal structure on . Let be a neighborhood of the diagonal in . Then the main result of the paper says:
1. There is a locally constructed -local Hodge decomposition ;
2. Let be a -local Hodge decomposition and with kernel . Then the measure is a -local Alexander Spanier cycle of dimension ;
3. The homology class of among -local cycles, , is independent of the choice of ;
4. The homology class of is equal to , where is the Hirzebruch-Thom -class and .