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Quasiconformal 4-manifolds. (English) Zbl 0704.57008

The authors deduce two fundamental results which clarify from a new viewpoint that 4-dimensional manifolds differ essentially from manifolds in any other dimension. Recall first that every pseudogroup of homeomorphisms of Euclidean space defines the corresponding category of manifolds. A homeomorphism $\phi$ with domain $D\subset {ℝ}^{n}$ is called quasiconformal if for all x in D

$\underset{r\to 0}{lim sup}\frac{max\left\{|\phi \left(y\right)-\phi \left(x\right)|||y-x|=r\right\}}{min\left\{|\phi \left(y\right)-\phi \left(x\right)|||y-x|=r\right\}}\phantom{\rule{1.em}{0ex}}\le \phantom{\rule{1.em}{0ex}}K$

with some $K\ge 1$. Hence the category of quasiconformal manifolds is intermediate between the topological manifolds and the smooth manifolds. The second author deduced [Geometric topology, Proc. Conf., Athens/Ga. 1977, 543-555 (1979; Zbl 0478.57007)] that for $n\ne 4$ any topological n-manifold admits a quasiconformal structure. Moreover, any two quasiconformal structures are equivalent by a homeomorphism isotopic to the identity. But for 4-dimensional manifolds the following two results are proved in the present paper. I. There are topological 4-manifolds which do not admit any quasiconformal structure. II. There are quasiconformal (indeed smooth) 4-manifolds which are homeomorphic but not quasiconformally equivalent. The proofs are presented in detail.

Reviewer: I.Kolář

##### MSC:
 57N13 Topology of the Euclidean 4-space, 4-manifolds 58H05 Pseudogroups and differentiable groupoids on manifolds
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