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Exotic knottings of surfaces in the 4-sphere. (English) Zbl 0635.57008

The authors announce the following theorem: Let Q be the connected sum of 10 copies of \({\mathbb{R}}P^ 2\). There exists an infinite series \(S_ 1,S_ 2,..\). of smooth submanifolds of \(S^ 4\) such that: (1) each \(S_ i\) is homeomorphic to Q, \(\pi_ 1(S^ 4\setminus S_ i)={\mathbb{Z}}/2{\mathbb{Z}}\), the normal Euler number (with local coefficients) of \(S_ i\) in \(S^ 4\) is 16; (2) for any \(i\neq j\) the pairs \((S^ 4,S_ i)\) and \((S^ 4,S_ j)\) are homeomorphic but not diffeomorphic.
The construction of the surfaces outlined in the paper goes by knotting the standard embedding \(Q\subset S^ 4\). The construction produces for every odd q a surface \(F_ q\subset S^ 4\) such that \(F_ q\approx Q\), \(\pi_ 1(S^ 4\setminus F_ q)={\mathbb{Z}}/2{\mathbb{Z}}\), the normal Euler number of \(F_ q\) is 16, and the 2-fold covering of \(S^ 4\) branched along \(F_ q\) is diffeomorphic to the Dolgachev complex elliptic surface \(D_{2,q}.\)
Recently, S. Donaldson, C. Okonek and A. van de Ven, R. Friedman and J. Morgan have proved that \(D_{2,q}\) is not diffeomorphic to \(D_{2,r}\) for odd \(q\neq r\). Thus, \((S^ 4,F_ q)\) is not diffeomorphic to \((S^ 4,F_ r)\) for \(q\neq r\). Existence of homeomorphisms stated in the Theorem uses surgery, applicable in dimension 4 by Friedman’s results. The obstructions for the pairs \((S^ 4,F_ q),(S^ 4,F_ r)\) to be homeomorphic are not known to be trivial but they sit in finite groups. This is sufficient for the existence of the desired subseries \(S_ 1,S_ 2,..\). of the series \(\{F_ q\}\).
Reviewer: V.Turaev

MSC:

57N13 Topology of the Euclidean \(4\)-space, \(4\)-manifolds (MSC2010)
57N35 Embeddings and immersions in topological manifolds
57R40 Embeddings in differential topology
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