New examples of Neuwirth–Stallings pairs and non-trivial real Milnor fibrations
[Nouveaux exemples de paires de Neuwirth–Stallings et fibrations de Milnor réelles non-triviales]
Annales de l'Institut Fourier, Tome 66 (2016) no. 1, pp. 83-104.

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.

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.

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DOI : 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
Mot clés : paire de Neuwirth–Stallings, structure de livre ouvert supérieure, espaces de configurations, fibre de Milnor réelle, germe d’application polynomiale.
Araújo dos Santos, Raimundo 1 ; Hohlenwerger, Maria A.B. 2, 3 ; Saeki, Osamu 4 ; Souza, Taciana O. 5

1 Universidade de São Paulo Instituto de Ciências Matemáticas e de Computação Av. Trabalhador São Carlense, 400, Centro, Postal Box 668, 13560-970 São Carlos, SP (Brazil)
2 Universidade de São Paulo Instituto de Ciências Matemáticas e de Computação Av. Trabalhador São Carlense, 400, Centro Postal Box 668, 13560-970, São Carlos, SP (Brazil)
3 and Universidade Federal do Recôncavo da Bahia Centro de Ciências Exatas e Tecnológicas Rua Rui Barbosa, 710, Centro, 44380-000 Cruz das Almas, BA (Brazil)
4 Kyushu University Institute of Mathematics for Industry Motooka 744, Nishi-ku Fukuoka 819-0395 (Japan)
5 Universidade Federal de Uberlândia Faculdade de Matemática Campus Santa Mônica - Bloco 1F - Sala 1F120 Av. João Naves de Avila, 2121, 38408-100 Uberlândia, MG (Brazil)
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     title = {New examples of {Neuwirth{\textendash}Stallings} pairs and non-trivial real {Milnor} fibrations},
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Araújo dos Santos, Raimundo; Hohlenwerger, Maria A.B.; Saeki, Osamu; Souza, Taciana O. New examples of Neuwirth–Stallings pairs and non-trivial real Milnor fibrations. Annales de l'Institut Fourier, Tome 66 (2016) no. 1, pp. 83-104. doi : 10.5802/aif.3006. http://www.numdam.org/articles/10.5802/aif.3006/

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