New examples of Neuwirth–Stallings pairs and non-trivial real Milnor fibrations

Araújo dos Santos, Raimundo; Hohlenwerger, Maria A.B.; Saeki, Osamu; Souza, Taciana O.

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 $({ℝ}^6,0)\to ({ℝ}^3,0)$ 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 $({ℝ}^{2p},0)\to ({ℝ}^p,0)$ ou $({ℝ}^{2p+1},0)\to ({ℝ}^p,0)$ 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 $({ℝ}^n,0)\to ({ℝ}^p,0)$ 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 $({ℝ}^6,0)\to ({ℝ}^3,0)$ 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 $({ℝ}^{2n},0)\to ({ℝ}^n,0)$ or $({ℝ}^{2n+1},0)\to ({ℝ}^n,0)$, $n⩾3$, 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 $({ℝ}^n,0)\to ({ℝ}^p,0)$ 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.

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.

[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 $({\Gamma }_4=0)$, 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, $\mathrm{Diff}\left(S^3\right)\simeq \mathrm{O}\left(4\right)$, 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