Transactions of the American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Help
ISSN 1088-6850(e) ISSN 0002-9947(p)

Simplifying stable mappings into the plane from a global viewpoint

Author(s): Mahito Kobayashi; Osamu Saeki
Journal: Trans. Amer. Math. Soc. 348 (1996), 2607-2636.
MSC (1991): Primary 57R45; Secondary 57R35, 57M99
MathSciNet review: 1344209
Retrieve article in: PDF
This article is available free of charge

Abstract | References | Similar articles | Additional information

Abstract: Let $f : M \to \text {\bf R}^{2}$ be a $C^{\infty }$ stable map of an $n$ -dimensional manifold into the plane. The main purpose of this paper is to define a global surgery operation on $f$ which simplifies the configuration of the critical value set and which does not change the diffeomorphism type of the source manifold $M$ . For this purpose, we also study the quotient space $W_{f}$ of $f$ , which is the space of the connected components of the fibers of $f$ , and we completely determine its local structure for arbitrary dimension $n$ of the source manifold $M$ . This is a completion of the result of Kushner, Levine and Porto for dimension 3 and that of Furuya for orientable manifolds of dimension 4. We also pay special attention to dimension 4 and obtain a simplification theorem for stable maps whose regular fiber is a torus or a 2-sphere, which is a refinement of a result of Kobayashi.

References:

[BdR]: O. Burlet and G. de Rham, Sur certaines applications génériques d'une variété close à 3 dimensions dans le plan, l'Enseign. Math. 20 (1974), 275--292. MR 51:1846
[C1]: J. Cerf, Sur les difféomorphismes de la sphère de dimension trois $(\Gamma _{4} = 0)$ , Lecture Notes in Math. 53, Springer, Berlin, 1968. MR 37:4824
[C2]: J. Cerf, La stratification naturelle des espaces de fonctions différentiables réelles et le théorème de la pseudo-isotopie, Publ. Math. Inst. Hautes Etudes Sci. 39 (1970), 5--173. MR 45:1176
[D]: S. K. Donaldson, Polynomial invariants for smooth four-manifolds, Topology 29 (1990), 257--315. MR 92a:57035
[Fr]: M. Freedman, The topology of four-dimensional manifolds, J. Diff. Geom 17 (1982), 357--453. MR 84b:57006
[Fu]: Y. Furuya, Sobre aplicações genéricas $M^{4} \to \text {\bf R}^{2}$ , Tese, ICMSC-USP, 1986.
[Gl]: H. Gluck, The embeddings of two-spheres in the four-sphere, Trans. Amer. Math. Soc. 104 (1962), 308--333. MR 27:1924
[GG]: M. Golubitsky and V. Guillemin, Stable mappings and their singularities, Grad. Texts in Math. 14, Springer, 1973. MR 49:6269
[H]: A. E. Hatcher, A proof of the Smale conjecture, $\text {\rm Diff}(S^{3}) \simeq O(4)$ , Ann. of Math. 117 (1983), 553--607. MR 85c:57008
[KM]: M. Kervaire and J. Milnor, Groups of homotopy spheres: I, Ann. of Math. 77 (1963), 504--537. MR 26:5584
[Kob1]: M. Kobayashi, Simplifying certain mappings from simply connected 4-manifolds into the plane, Tokyo J. Math. 15 (1992), 327--349. MR 93k:57062
[Kob2]: M. Kobayashi, Stable maps with trivial monodromies and inactive log-transformations, preprint, Tokyo Institute of Technology, November 1993.
[Kot1]: D. Kotschick, Orientation-reversing homeomorphism in surface geography, Math. Ann. 292 (1992), 375--381. MR 93f:14019
[Kot2]: D. Kotschick, Non-trivial harmonic spinors on certain algebraic surfaces, Einstein metrics and Yang-Mills connections (ed. T. Mabuchi), Lecture Notes in Pure and Appl. Math., vol. 145, Dekker, New York, 1993, pp. 85--88. MR 94d:58138
[KLP]: L. Kushner, H. Levine and P. Porto, Mapping three-manifolds into the plane I, Bol. Soc. Mat. Mexicana 29 (1984), 11--33. MR 86j:58011
[L1]: H. Levine, Elimination of cusps, Topology 3 (suppl. 2) (1965), 263--296. MR 31:756
[L2]: H. Levine, Classifying immersions into $\text {\bf R}^{4}$ over stable maps of 3-manifolds into $\text {\bf R}^{2}$ , Lect. Notes in Math., vol. 1157, Springer-Verlag, Berlin-Heidelberg-New York-Tokyo, 1985. MR 88f:57056
[ML]: L. E. Mata-Lorenzo, Polyhedrons and stable maps from 3-manifolds into the plane, unpublished note.
[Ma]: J. Mather, Stability of $C^{\infty }$ mappings: VI. The nice dimensions, Proc. Liverpool Singularities - Symposium I, Lecture Notes in Math., vol. 192, Springer-Verlag, Berlin-Heidelberg-New York, 1971, pp. 207--253. MR 45:2747
[Mi]: J. Milnor, Lectures on the -cobordism theorem, Princeton Univ. Press, Princeton, 1965. MR 32:8352
[MT1]: B. Moishezon and M. Teicher, Existence of simply connected algebraic surfaces of general type with positive and zero indices, Proc. Nat. Acad. Sci. USA 83 (1986), 6665--6666. MR 87h:14034
[MT2]: B. Moishezon and M. Teicher, Simply-connected algebraic surfaces of positive index, Inv. Math. 89 (1987), 601--643. MR 88f:14037
[MPS]: W. Motta, P. Porto and O. Saeki, Stable maps of 3-manifolds into the plane and their quotient spaces, Proc. London Math. Soc., vol. 71, 1995, pp. 158--174.
[PF]: P. Porto and Y. Furuya, On special generic maps from a closed manifold into the plane, Topology Appl. 35 (1990), 41--52. MR 91c:57034
[Sa1]: O. Saeki, Topology of special generic maps of manifolds into Euclidean spaces, Topology Appl. 49 (1993), 265--293. MR 93m:57031
[Sa2]: O. Saeki, Simple stable maps of 3-manifolds into surfaces II, J. Fac. Sci. Univ. Tokyo 40 (1993), 73--124. MR 94k:57044
[Sm]: S. Smale, Diffeomorphisms of the 2-sphere, Proc. Amer. Math. Soc. 10 (1959), 621--626. MR 22:3004

	Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
\|