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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 ϕ with domain D n is called quasiconformal if for all x in D

lim sup r0 max{|ϕ(y)-ϕ(x)|||y-x|=r} min{|ϕ(y)-ϕ(x)|||y-x|=r}K

with some K1. 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 n4 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:
57N13Topology of the Euclidean 4-space, 4-manifolds
58H05Pseudogroups and differentiable groupoids on manifolds
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