New examples of Neuwirth–Stallings pairs and non-trivial real Milnor fibrations
Annales de l'Institut Fourier, Volume 66 (2016) no. 1, p. 83-104
We use the topology of configuration spaces to give a characterization of Neuwirth–Stallings pairs (S 5 ,K) with dimK=2. As a consequence, we construct polynomial map germs ( 6 ,0)( 3 ,0) with an isolated singularity at the origin such that their Milnor fibers are not diffeomorphic to a disk, thus putting an end to Milnor’s non-triviality question. Furthermore, for a polynomial map germ ( 2n ,0)( n ,0) or ( 2n+1 ,0)( n ,0), n3, with an isolated singularity at the origin, we study the conditions under which the associated Milnor fiber has the homotopy type of a bouquet of spheres. We then construct, for every pair (n,p) with n/2p2, a new example of a polynomial map germ ( n ,0)( p ,0) with an isolated singularity at the origin such that its Milnor fiber has the homotopy type of a bouquet of a positive number of spheres.
Nous utilisons la topologie des espaces de configuration pour caractériser les paires de Neuwirth–Stallings (S 5 ,K), où K est de dimension 2. En conséquence, nous construisons des germes d’applications polynomiales ( 6 ,0)( 3 ,0) ayant une singularité isolée à l’origine tels que leurs fibres de Milnor ne soient pas difféomorphes au disque, mettant ainsi un terme à la question de non-trivialité due à Milnor. En outre, pour un germe d’application polynomiale ( 2p ,0)( p ,0) ou ( 2p+1 ,0)( p ,0) ayant une singularité isolée à l’origine, nous étudions les conditions dans lesquelles la fibre de Milnor associée ait le type d’homotopie d’un bouquet de sphères. De plus, nous construisons pour chaque paire (n,p), où n/2p2, un nouveau exemple d’un germe d’application polynomiale ( n ,0)( p ,0) ayant une singularité isolée à l’origine tel que la fibre de Milnor associée ait le type d’homotopie d’un bouquet de sphères non triviales.
Received : 2014-08-26
Revised : 2015-03-11
Accepted : 2015-03-26
Published online : 2016-02-17
DOI : https://doi.org/10.5802/aif.3006
Classification:  32S55,  57R45,  58K05
Keywords: Neuwirth–Stallings pair, higher open book structure, configuration space, real Milnor fiber, real polynomial map germ
@article{AIF_2016__66_1_83_0,
     author = {Ara\'ujo dos Santos, Raimundo and Hohlenwerger, Maria A.B. and Saeki, Osamu and Souza, Taciana O.},
     title = {New examples of Neuwirth--Stallings pairs and non-trivial real Milnor fibrations},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {66},
     number = {1},
     year = {2016},
     pages = {83-104},
     doi = {10.5802/aif.3006},
     language = {en},
     url = {https://aif.centre-mersenne.org/item/AIF_2016__66_1_83_0}
}
New examples of Neuwirth–Stallings pairs and non-trivial real Milnor fibrations. Annales de l'Institut Fourier, Volume 66 (2016) no. 1, pp. 83-104. doi : 10.5802/aif.3006. https://aif.centre-mersenne.org/item/AIF_2016__66_1_83_0/

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