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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 \(D\subset {\mathbb{R}}^ n\) is called quasiconformal if for all x in D \[ \limsup_{r\to 0}\frac{\max \{| \phi (y)-\phi (x)| | | y-x| =r\}}{\min \{| \phi (y)-\phi (x)| | | y-x| =r\}}\quad \leq \quad K \] with some \(K\geq 1\). 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 \(n\neq 4\) 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 (MSC2010)
58H05 Pseudogroups and differentiable groupoids

Citations:

Zbl 0478.57007
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References:

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