Singular fibers of stable maps and signatures of 4–manifolds

Saeki, Osamu¹ ; Yamamoto, Takahiro²

¹ Faculty of Mathematics, Kyushu University, Hakozaki, Fukuoka 812-8581, Japan
² Department of Mathematics, Hokkaido University, Sapporo 060-0810, Japan

Geometry & topology, Tome 10 (2006) no. 1, pp. 359-399.

Résumé

We show that for a $C^{\infty }$ stable map of an oriented 4–manifold into a 3–manifold, the algebraic number of singular fibers of a specific type coincides with the signature of the source 4–manifold.

Détail

DOI : 10.2140/gt.2006.10.359

Keywords: stable map, singular fiber, chiral fiber, signature, universal complex, bordism

Bibliographie

[1] Y Ando, On local structures of the singularities Ak, Dk and Ek of smooth maps, Trans. Amer. Math. Soc. 331 (1992) 639 | DOI

[2] T Bröcker, K Jänich, Introduction to differential topology, Cambridge University Press (1982)

[3] P E Conner, E E Floyd, Differentiable periodic maps, 33, Springer (1964)

[4] J Damon, Topological properties of real simple germs, curves, and the nice dimensions n > p, Math. Proc. Cambridge Philos. Soc. 89 (1981) 457 | DOI

[5] C Ehresmann, Sur les espaces fibrés différentiables, C. R. Acad. Sci. Paris 224 (1947) 1611

[6] H Endo, Meyer’s signature cocycle and hyperelliptic fibrations, Math. Ann. 316 (2000) 237 | DOI

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

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

[9] M È Kazaryan, Hidden singularities and Vassiliev’s homology complex of singularity classes, Mat. Sb. 186 (1995) 119 | DOI

[10] L Kushner, H Levine, P Porto, Mapping three-manifolds into the plane I, Bol. Soc. Mat. Mexicana 29 (1984) 11

[11] K Lamotke, The topology of complex projective varieties after S Lefschetz, Topology 20 (1981) 15 | DOI

[12] H Levine, Classifying immersions into R4 over stable maps of 3-manifolds into R2, 1157, Springer (1985)

[13] J N Mather, Stability of C∞ mappings V : Transversality, Advances in Math. 4 (1970)

[14] J N Mather, Stability of C∞ mappings VI : The nice dimensions, from: "Proceedings of Liverpool Singularities Symposium I (1969/70)", Lecture Notes in Mathematics 192, Springer (1971) 207

[15] Y Matsumoto, On 4–manifolds fibered by tori II, Proc. Japan Acad. Ser. A Math. Sci. 59 (1983) 100

[16] Y Matsumoto, Lefschetz fibrations of genus two – a topological approach, from: "Topology and Teichmüller spaces (Katinkulta, 1995)", World Sci. Publishing (1996) 123

[17] W Meyer, Die Signatur von Flächenbündeln, Math. Ann. 201 (1973) 239 | DOI

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

[19] T Ohmoto, O Saeki, K Sakuma, Self-intersection class for singularities and its application to fold maps, Trans. Amer. Math. Soc. 355 (2003) 3825 | DOI

[20] R Rimányi, A Szücs, Pontrjagin–Thom-type construction for maps with singularities, Topology 37 (1998) 1177 | DOI

[21] R Sadykov, Elimination of singularities of smooth mappings of 4–manifolds into 3–manifolds, Topology Appl. 144 (2004) 173 | DOI

[22] O Saeki, Topology of special generic maps of manifolds into Euclidean spaces, Topology Appl. 49 (1993) 265 | DOI

[23] O Saeki, Fold maps on 4–manifolds, Comment. Math. Helv. 78 (2003) 627 | DOI

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

[25] K Sakuma, On special generic maps of simply connected 2n–manifolds into R3, Topology Appl. 50 (1993) 249 | DOI

[26] V A Vassiliev, Lagrange and Legendre characteristic classes, 3, Gordon and Breach Science Publishers (1988)

[27] T Yamamoto, Classification of singular fibres of stable maps of 4–manifolds into 3–manifolds and its applications, J. Math. Soc. Japan (to appear)

[28] T Yamamoto, Classification of singular fibers and its applications, dissertation, Hokkaido University (2002)