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Zbl 0704.57008
Donaldson, S.K.; Sullivan, D.P.
Quasiconformal 4-manifolds. (English)
[J] Acta Math. 163, No.3-4, 181-252 (1989). ISSN 0001-5962; ISSN 1871-2509/e

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 $D\subset {\bbfR}\sp n$ is called quasiconformal if for all x in D $$ \limsup\sb{r\to 0}\frac{\max \{\vert \phi (y)-\phi (x)\vert \vert \vert y-x\vert =r\}}{\min \{\vert \phi (y)-\phi (x)\vert \vert \vert y-x\vert =r\}}\quad \le \quad K $$ 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.
[I.Kolář]

MSC 2000:: *57N13 Topology of Euclidean 4-space, 4-manifolds
58H05 Pseudogroups on manifolds

Keywords: topological 4-manifolds without quasiconformal structure; 4-dimensional manifolds; pseudogroup of homeomorphisms; quasiconformal manifolds; homeomorphic but not quasiconformally equivalent

Citations: Zbl 0478.57007

Cited in: Zbl 0843.30020 Zbl 1001.57502 Zbl 0785.30008 Zbl 0785.30009

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