# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
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 ${ℝ}^{n}$ is quasiconformal if there is a $K>0$ such that for each $x$

$\underset{r\to 0}{\overline{lim}}\frac{max\left\{\left|h\left(x\right)-h\left(y\right)\right|;|x-y|=r\right\}}{min\left\{\left|h\left(x\right)-h\left(y\right)\right|;|x-y|=r\right\}}:=K\left(x\right)

Let $M$ be a compact oriented quasiconformal manifold of even dimension $2l$. Let $\gamma$ be the $ℤ/2$-grading of ${L}^{2}\left(M,{\wedge }^{l}{T}_{ℂ}^{*}\right)$ 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 $L=H\gamma H+\gamma$ with kernel $L\left(x,y\right)$. Then the measure $\sigma =\text{tr}\left({\wedge }^{2q+1}L\right)$ is a ${U}^{2q}$-local Alexander Spanier cycle of dimension $2q$;

3. The homology class of $\sigma$ among ${U}^{r}$-local cycles, $r=2q\left(6l+2\right)$, is independent of the choice of $H$;

4. The homology class of $\sigma$ is equal to ${\lambda }_{2q}\left({L}_{2l-2q}\cap \left[M\right]\right)$, where $L$ is the Hirzebruch-Thom $L$-class and ${\lambda }_{2q}={2}^{2q+1}{\left(2\pi i\right)}^{-q}q!/2q!$.

Reviewer: W.Lück (Mainz)

##### MSC:
 57N99 Topological manifolds