no. 1
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]
Araújo dos Santos, Raimundo1 ; Hohlenwerger, Maria A.B.2, 3 ; Saeki, Osamu4 ; 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)
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 (ℝ 6 ,0)→(ℝ 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 (ℝ 2p ,0)→(ℝ p ,0) ou (ℝ 2p+1 ,0)→(ℝ 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/2⩾p⩾2, un nouveau exemple d’un germe d’application polynomiale (ℝ n ,0)→(ℝ 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 (ℝ 6 ,0)→(ℝ 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 (ℝ 2n ,0)→(ℝ n ,0) or (ℝ 2n+1 ,0)→(ℝ n ,0), n⩾3, 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/2⩾p⩾2, a new example of a polynomial map germ (ℝ n ,0)→(ℝ 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.

Reçu le :
Révisé le :
Accepté le :
Publié le :
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) 
@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{\textendash}Stallings} pairs and non-trivial real {Milnor} fibrations},
     journal = {Annales de l'Institut Fourier},
     pages = {83--104},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {66},
     number = {1},
     year = {2016},
     doi = {10.5802/aif.3006},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3006/}
}
TY  - JOUR
AU  - Araújo dos Santos, Raimundo
AU  - Hohlenwerger, Maria A.B.
AU  - Saeki, Osamu
AU  - Souza, Taciana O.
TI  - New examples of Neuwirth–Stallings pairs and non-trivial real Milnor fibrations
JO  - Annales de l'Institut Fourier
PY  - 2016
SP  - 83
EP  - 104
VL  - 66
IS  - 1
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.3006/
DO  - 10.5802/aif.3006
LA  - en
ID  - AIF_2016__66_1_83_0
ER  - 
%0 Journal Article
%A Araújo dos Santos, Raimundo
%A Hohlenwerger, Maria A.B.
%A Saeki, Osamu
%A Souza, Taciana O.
%T New examples of Neuwirth–Stallings pairs and non-trivial real Milnor fibrations
%J Annales de l'Institut Fourier
%D 2016
%P 83-104
%V 66
%N 1
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.3006/
%R 10.5802/aif.3006
%G en
%F AIF_2016__66_1_83_0
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. https://aif.centre-mersenne.org/articles/10.5802/aif.3006/

[1] Artin, Emil Zur Isotopie zweidimensionaler Flächen im R 4 , Abh. Math. Sem. Univ. Hamburg, Volume 4 (1925) no. 1, pp. 174-177 | DOI

[2] Cerf, Jean Sur les difféomorphismes de la sphère de dimension trois (Γ 4 =0), Lecture Notes in Mathematics, No. 53, Springer-Verlag, Berlin-New York, 1968, xii+133 pages

[3] Church, P. T.; Lamotke, K. Non-trivial polynomial isolated singularities, Nederl. Akad. Wetensch. Proc. Ser. A 78=Indag. Math., Volume 37 (1975), pp. 149-154

[4] Dutertre, Nicolas; Araújo dos Santos, Raimundo N Topology of real Milnor fibration for non-isolated singularities (http://arxiv.org/abs/1211.6233, 2012)

[5] Fadell, Edward R.; Husseini, Sufian Y. Geometry and topology of configuration spaces, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2001, xvi+313 pages | DOI

[6] Funar, Louis Global classification of isolated singularities in dimensions (4,3) and (8,5), Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), Volume 10 (2011) no. 4, pp. 819-861

[7] Haefliger, André Differentiable links, Topology, Volume 1 (1962) no. 3, pp. 241-244

[8] Hatcher, Allen E. A proof of the Smale conjecture, Diff (S 3 )≃O(4), Ann. of Math. (2), Volume 117 (1983) no. 3, pp. 553-607 | DOI

[9] Kervaire, Michel A.; Milnor, John W. Groups of homotopy spheres. I, Ann. of Math. (2), Volume 77 (1963) no. 3, pp. 504-537

[10] Khimshiashvili, G. N. The local degree of a smooth mapping, Sakharth. SSR Mecn. Akad. Moambe, Volume 85 (1977) no. 2, pp. 309-312

[11] Looijenga, Eduard A note on polynomial isolated singularities, Nederl. Akad. Wetensch. Proc. Ser. A 74=Indag. Math., Volume 33 (1971), pp. 418-421

[12] Milnor, John Singular points of complex hypersurfaces, Annals of Mathematics Studies, No. 61, Princeton University Press, Princeton, N.J., 1968, iii+122 pages

[13] Milnor, John Differential topology forty-six years later, Notices Amer. Math. Soc., Volume 58 (2011), pp. 804-809

[14] Palais, Richard S. Local triviality of the restriction map for embeddings, Comment. Math. Helv., Volume 34 (1960) no. 1, pp. 305-312

[15] Araújo dos Santos, R.; Dreibelbis, D.; Dutertre, N. Topology of the real Milnor fiber for isolated singularities, Real and complex singularities (Contemp. Math.), Volume 569, Amer. Math. Soc., Providence, RI, 2012, pp. 67-75 | DOI

[16] Araújo dos Santos, Raimundo; Tibăr, Mihai Real map germs and higher open book structures, Geom. Dedicata, Volume 147 (2010), pp. 177-185 | DOI

[17] Araújo dos Santos, Raimundo N.; Chen, Ying; Tibăr, Mihai Singular open book structures from real mappings, Cent. Eur. J. Math., Volume 11 (2013) no. 5, pp. 817-828 | DOI

[18] Araújo dos Santos, Raimundo N.; Chen, Ying; Tibăr, Mihai Real polynomial maps and singular open books at infinity (2014) (http://arxiv.org/abs/1401.8286)

[19] Smale, Stephen Generalized Poincaré’s conjecture in dimensions greater than four, Ann. of Math. (2), Volume 74 (1961) no. 2, pp. 391-406

[20] Souza, T. O.; Hohlenwerger, M. A. B.; De Mattos, D.; Araújo Dos Santos, R. New characterization of trivial maps in 3-dimensional real Milnor fibers, JP J. Geom. Topol., Volume 12 (2012) no. 2, pp. 207-217

[21] Zeeman, E. C. Twisting spun knots, Trans. Amer. Math. Soc., Volume 115 (1965), pp. 471-495

Cité par Sources :