Global classification of isolated singularities in dimensions (4,3) and (8,5)

Funar, Louis

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 10 (2011) no. 4, pp. 819-861.

We characterize those closed $2k$-manifolds admitting smooth maps into $(k+1)$-manifolds with only finitely many critical points, for $k\in \{2,4\}$. We compute then the minimal number of critical points of such smooth maps for $k=2$ and, under some fundamental group restrictions, also for $k=4$. The main ingredients are King’s local classification of isolated singularities, decomposition theory, low dimensional cobordisms of spherical fibrations and 3-manifolds topology.

Classification : 57R45, 58K05, 57R60, 57R70

[1] D. Andrica and L. Funar, On smooth maps with finitely many critical points, J. London Math. Soc. 69 (2004), 783–800, Addendum 73 (2006), 231–236. | MR | Zbl

[2] D. Andrica, L. Funar and E. Kudryavtseva, On the minimal number of critical points of maps between closed manifolds, Russian Journal of Mathematical Physics, special issue “Conference for the 65-th birthday of Nicolae Teleman”, J.-P. Brasselet, A. Legrand, R. Longo, A. Mishchenko (Eds.), 16 (2009), 363–370. | MR | Zbl

[3] P. L. Antonelli, Structure theory for Montgomery-Samelson between manifolds, I, II Canad. J. Math. 21 (1969), 170–179, 180–186. | MR | Zbl

[4] P. L. Antonelli, Differentiable Montgomery-Samelson fiberings with finite singular sets, Canad. J. Math. 21 (1969), 1489–1495. | MR | Zbl

[5] R. Araújo dos Santos and M. Tibăr, Real map germs and higher open books, Geom. Dedicata 147 (2010), 177–185. | MR | Zbl

[6] E. Artin and R. Fox, Some wild cells and spheres in three-dimensional space, Ann. of Math. 49 (1948), 979–990. | MR | Zbl

[7] L. Bessières, G. Besson, M. Boileau, S. Maillot and J. Porti, “Geometrisation of 3-Manifolds”, EMS Tracts Math., Vol. 13, Zurich, 2010. | MR | Zbl

[8] E. M. Brown, Unknotted solid tori and genus one Whitehead manifolds, Trans. Amer. Math. Soc. 333 (1992), 835–847. | MR | Zbl

[9] D. Burghelea, R. Lashof and M. Rothenberg, “Groups of Automorphisms of Manifolds”, Lecture Notes in Mathematics, Vol. 473, Springer-Verlag, 1975. | MR | Zbl

[10] J. Cerf, “Sur les difféomorphismes de la sphère de dimension trois $({\Gamma }_4=0)$”, Lecture Notes in Mathematics, Vol. 53, Springer, Berlin, 1968. | MR | Zbl

[11] P. T. Church and K. Lamotke, Non-trivial polynomial isolated singularities, Indag. Math. 37 (1975), 149–154. | MR | Zbl

[12] P. T. Church and J. G. Timourian, Differentiable maps with $0$-dimensional critical set I, Pacific J. Math. 41 (1972), 615–630. | MR | Zbl

[13] P. T. Church and J. G. Timourian, Continuous maps with $0$-dimensional branch set, Indiana Univ. Math. J. 23 (1973/1974), 949–958. | MR | Zbl

[14] P. T. Church and J. G. Timourian, Differentiable maps with $0$-dimensional critical set II, Indiana Univ. Math. J. 24 (1974), 17–28. | MR | Zbl

[15] P. E. Conner, On the impossibility of fibring a certain manifold by a compact fibre, Michigan Math. J. 5 (1957), 249–255. | MR | Zbl

[16] R. Daverman, “Decompositions of Manifolds”, Academic Press, 1986. | MR | Zbl

[17] A. Dimca, “Singularities and Topology of Hypersurfaces”, Springer-Verlag, Berlin, 1992. | MR | Zbl

[18] A. Dold and H. Whitney, Classification of oriented bundles over a 4-complex, Ann. of Math. 69 (1959), 667–677. | MR | Zbl

[19] C. Earle and J. Eells Jr., A fibre bundle description of Teichmüller theory, J. Differential Geom. 3 (1969), 19–43. | MR | Zbl

[20] C. Earle and J. H. Schatz, Teichmüller theory for surfaces with boundary, J. Differential Geom. 4 (1970), 169–185. | MR | Zbl

[21] E. R. Fadell and S. Y. Husseini, “Geometry and Topology of Configuration Spaces”, Springer Monograph Math., 2001. | MR | Zbl

[22] L. Funar, C. Pintea and P. Zhang, Examples of smooth maps with finitely many critical points in dimensions $(4,3)$, $(8,5)$ and $(16,9)$, Proc. Amer. Math. Soc. 138 (2010), 355–365. | MR | Zbl

[23] A. Haefliger, Differentiable imbeddings, Bull. Amer. Math. Soc. 67 (1961), 109–112. | MR | Zbl

[24] A. Haefliger, Plongements de variétés dans le domaine stable, In: “Séminaire Bourbaki 1962/1963”, Vol. 8, Exp. No. 245, 63–77, Soc. Math. France, Paris, 1995. | EuDML | Numdam | MR | Zbl

[25] A. Haefliger, Plongements différentiables dans le domaine stable, Comment. Math. Helv. 37 (1962/1963), 155–176. | EuDML | MR | Zbl

[26] A. Haefliger, Differentiable embeddings of $S^n$ in $S^{n+q}$ for $q>2$, Ann. of Math. 83 (1966), 402–436. | MR | Zbl

[27] M.-E. Hamstrom, Regular mappings and the space of homeomorphisms on a $3$-manifold, Mem. Amer. Math. Soc. 40 (1961), 42. | MR | Zbl

[28] A. Hatcher, Homeomorphisms of sufficiently large $P^2$-irreducible $3$-manifolds, Topology 15 (1976), 343–347. | MR | Zbl

[29] A. Hatcher, A proof of a Smale conjecture, $\mathrm{Diff}\left(S^3\right)\simeq \mathrm{O}\left(4\right)$, Ann. of Math. (2) 117 (1983), 553–607. | MR | Zbl

[30] A. Hatcher and D. McCullough, Finiteness of classifying spaces of relative diffeomorphism groups of $3$-manifolds, Geom. Topol. 1 (1997), 91–109. | EuDML | MR | Zbl

[31] J. Hempel, “3-manifolds”, reprint of the 1976 original, AMS Chelsea Publishing, 2004. | MR | Zbl

[32] W. Huebsch and M. Morse, Schoenflies extensions without interior differential singularities, Ann. of Math. 76 (1962), 18–54. | MR | Zbl

[33] W. Jaco, Three manifolds with fundamental group a free product, Bull. Amer. Math. Soc. 75 (1969), 972–977. | MR | Zbl

[34] I. M. James and J. H. C. Whithead, The homotopy theory of sphere bundles over spheres. II, Proc. London Math. Soc. (3) 5 (1955), 148–166. | MR | Zbl

[35] L. Kauffman and W. Neumann, Products of knots, branched fibrations and sums of singularities, Topology 16 (1977), 369–393. | MR | Zbl

[36] H. C. King, Topological type of isolated singularities, Ann. of Math. 107 (1978), 385–397. | MR | Zbl

[37] H. C. King, Topology of isolated critical points of functions on singular spaces, In: “Stratifications, Singularities and Differential Equations”, II (Marseille, 1990; Honolulu, HI, 1990), 63–72, David Trotman and Leslie Charles Wilson (eds.), Travaux en Cours, 55, Hermann, Paris, 1997. | MR | Zbl

[38] A. Kosinski, On the inertia group of $\pi$- manifolds, Amer. J. Math. 89 (1967), 227–248. | MR | Zbl

[39] A. Kosinski, “Differentiable Manifolds”, Academic Press, London, 1993. | Zbl

[40] E. Looijenga, A note on polynomial isolated singularities, Indag. Math. 33 (1971), 418–421. | MR | Zbl

[41] J. Milnor, On manifolds homeomorphic to the $7$-sphere, Ann. of Math. (2) 64 (1956), 399–405. | MR | Zbl

[42] J. Milnor, Differentiable structures on spheres, Amer. J. Math. 81 (1959), 962–972. | MR | Zbl

[43] J. Milnor, A unique decomposition theorem for $3$-manifolds, Amer. J. Math. 84 (1962), 1–7. | MR | Zbl

[44] J. Milnor, “Singular Points of Complex Hypersurfaces”, Princeton University Press, 1968. | MR | Zbl

[45] D. Montgomery and H. Samelson, Fiberings with singularities, Duke Math. J. 13 (1946), 51–56. | MR | Zbl

[46] P. E. Pushkar and Yu. B. Rudyak, On the minimal number of critical points of functions on $h$-cobordisms, Math. Res. Lett. 9 (2002), 241–246. | MR | Zbl

[47] E. G. Rees, On a question of Milnor concerning singularities of maps, Proc. Edinburgh Math. Soc. (2) 43 (2000), 149–153. | MR | Zbl

[48] D. Rolfsen, “Knots and Links”, corrected reprint of the 1976 original, Mathematics Lecture Series, 7, Publish or Perish, Inc., 1990. | MR | Zbl

[49] L. Rudolph, Isolated critical points of mappings from ${\mathbf{R}}^4$ to ${\mathbf{R}}^2$ and a natural splitting of the Milnor number of a classical fibered link. I. Basic theory; examples, Comment. Math. Helv. 62 (1987), 630–645. | EuDML | MR | Zbl

[50] R. Schultz, On the inertia group of a product of spheres, Trans. Amer. Math. Soc. 156 (1971), 137–153. | MR | Zbl

[51] P. Scott, There are no fake Seifert fibre spaces with infinite ${\pi }_1$, Ann. of Math. (2) 117 (1983), 35–70. | MR | Zbl

[52] L. C. Siebenmann, Approximating cellular maps by homeomorphisms, Topology 11 (1972), 271–294. | MR | Zbl

[53] L. C. Siebenmann, Deformation of homeomorphisms of stratified sets, Comment. Math. Helv. 47 (1972), 123–163. | EuDML | MR | Zbl

[54] E. H. Spanier, “Algebraic Topology”, Springer-Verlag, New York-Berlin, 1981. | MR | Zbl

[55] N. Steenrod, “The Topology of Fibre Bundles”, reprint of the 1957 edition, Princeton University Press, 1999. | MR | Zbl

[56] F. Takens, Isolated critical points of $C^{\infty }$ and $C^{\omega }$ functions, Indag. Math. 29 (1967), 238–243. | MR | Zbl

[57] F. Takens, The minimal number of critical points of a function on a compact manifold and the Lusternik-Schnirelman category, Invent. Math. 6 (1968), 197–244. | EuDML | MR | Zbl

[58] J. G. Timourian, Fiber bundles with discrete singular set, J. Math. Mech. 18 (1968), 61–70. | MR | Zbl

[59] F. Waldhausen, On irreducible $3$-manifolds which are sufficiently large, Ann. of Math. (2) 87 (1968), 56–88. | MR | Zbl

[60] Shicheng Wang and Ying Qing Wu, Covering invariants and co-hopficity of $3$-manifold groups, Proc. London Math. Soc. (3) 68 (1994), 203–224. | MR | Zbl

[61] J. H. C. Whitehead, On finite cocycles and the sphere theorem, Colloq. Math. 6(1958), 271–281. | EuDML | MR | Zbl

[62] L. M. Woodward, The classification of orientable vector bundles over CW-complexes of small dimension, Proc. Roy. Soc. Edinburgh, 92A (1982), 175–179. | MR | Zbl

[63] Yu Fengchun and Shicheng Wang, Covering degrees are determined by graph manifolds involved, Comment. Math. Helv. 74 (1999), 238–247. | MR | Zbl