Singular fibers of stable maps of 3–manifolds with boundary into surfaces and their applications

Saeki, Osamu¹ ; Yamamoto, Takahiro²

¹Institute of Mathematics for Industry, Kyushu University, Motooka 744, Nishi-ku, Fukuoka 819-0395, Japan
²Faculty of Engineering, Kyushu Sangyo University, 3-1 Matsukadai 2-chome, Higashi-ku, Fukuoka 813-8503, Japan

Algebraic and Geometric Topology, Tome 16 (2016) no. 3, pp. 1379-1402

Résumé

We first classify singular fibers of proper C∞ stable maps of 3–dimensional manifolds with boundary into surfaces. Then we compute the cohomology groups of the associated universal complex of singular fibers, and obtain certain cobordism invariants for Morse functions on compact surfaces with boundary.

Détail

DOI : 10.2140/agt.2016.16.1379

Classification : 57R45, 57R35, 57R90, 58K15, 58K65
Keywords: stable map, singular fiber, manifold with boundary, cobordism

Bibliographie

[1] J Damon, The relation between C∞ and topological stability, Bol. Soc. Brasil. Mat. 8 (1977) 1

[2] C G Gibson, K Wirthmüller, A A Du Plessis, E J N Looijenga, Topological stability of smooth mappings, 552, Springer (1976)

[3] M Golubitsky, V Guillemin, Stable mappings and their singularities, 14, Springer (1973)

[4] L F Martins, A C Nabarro, Projections of hypersurfaces in R4 with boundary to planes, Glasg. Math. J. 56 (2014) 149

[5] T Ohmoto, Vassiliev complex for contact classes of real smooth map-germs, Rep. Fac. Sci. Kagoshima Univ. Math. Phys. Chem. 27 (1994) 1

[6] A Du Plessis, T Wall, The geometry of topological stability, 9, Clarendon Press (1995)

[7] O Saeki, Topology of singular fibers of differentiable maps, 1854, Springer (2004)

[8] O Saeki, Cobordism of Morse functions on surfaces, the universal complex of singular fibers and their application to map germs, Algebr. Geom. Topol. 6 (2006) 539

[9] O Saeki, S Takahashi, Visual data mining based on differential topology: a survey, Pac. J. Math. Ind. 6 (2014)

[10] O Saeki, S Takahashi, D Sakurai, H Y Wu, K Kikuchi, H Carr, D Duke, T Yamamoto, Visualizing multivariate data using singularity theory, from: "The impact of applications on mathematics" (editors M Wakayama, R S Anderssen, J Cheng, Y Fukumoto, R McKibbin, K Polthier, T Takagi, K C Toh), Math. Ind. (Tokyo) 1, Springer (2014) 51

[11] O Saeki, T Yamamoto, Singular fibers of stable maps and signatures of 4–manifolds, Geom. Topol. 10 (2006) 359

[12] O Saeki, T Yamamoto, Singular fibers and characteristic classes, Topology Appl. 155 (2007) 112

[13] O Saeki, T Yamamoto, Co-orientable singular fibers of stable maps of 3–manifolds with boundary into surfaces, Sūrikaisekikenkyūsho Kōkyūroku 1948 (2015) 137

[14] N Shibata, On non-singular stable maps of 3–manifolds with boundary into the plane, Hiroshima Math. J. 30 (2000) 415

[15] V A Vassilyev, Lagrange and Legendre characteristic classes, 3, Gordon and Breach (1988)

[16] T Yamamoto, Classification of singular fibres of stable maps of 4–manifolds into 3–manifolds and its applications, J. Math. Soc. Japan 58 (2006) 721

[17] T Yamamoto, Euler number formulas in terms of singular fibers of stable maps, from: "Real and complex singularities" (editors A Harris, T Fukui, S Koike), World Sci. (2007) 427

[18] T Yamamoto, Singular fibers of two-colored maps and cobordism invariants, Pacific J. Math. 234 (2008) 379