Global classification of isolated singularities in dimensions (4,3) and (8,5)
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 10 (2011) no. 4, p. 819-861

We characterize those closed 2k-manifolds admitting smooth maps into (k+1)-manifolds with only finitely many critical points, for k{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.

Published online : 2018-06-21
Classification:  57R45,  58K05,  57R60,  57R70
@article{ASNSP_2011_5_10_4_819_0,
     author = {Funar, Louis},
     title = {Global classification of isolated singularities in dimensions (4,3) and (8,5)},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 10},
     number = {4},
     year = {2011},
     pages = {819-861},
     zbl = {1241.57037},
     mrnumber = {2932895},
     language = {en},
     url = {http://www.numdam.org/item/ASNSP_2011_5_10_4_819_0}
}
Funar, Louis. Global classification of isolated singularities in dimensions (4,3) and (8,5). Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 10 (2011) no. 4, pp. 819-861. http://www.numdam.org/item/ASNSP_2011_5_10_4_819_0/

[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 2197380 | Zbl 1052.57051

[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 2551883 | Zbl 1194.58032

[3] P. L. Antonelli, Structure theory for Montgomery-Samelson between manifolds, I, II Canad. J. Math. 21 (1969), 170–179, 180–186. | MR 238320 | Zbl 0169.54901

[4] P. L. Antonelli, Differentiable Montgomery-Samelson fiberings with finite singular sets, Canad. J. Math. 21 (1969), 1489–1495. | MR 261624 | Zbl 0187.45101

[5] R. Araújo dos Santos and M. Tibăr, Real map germs and higher open books, Geom. Dedicata 147 (2010), 177–185. | MR 2660575 | Zbl 1202.32026

[6] E. Artin and R. Fox, Some wild cells and spheres in three-dimensional space, Ann. of Math. 49 (1948), 979–990. | MR 27512 | Zbl 0033.13602

[7] L. Bessières, G. Besson, M. Boileau, S. Maillot and J. Porti, “Geometrisation of 3-Manifolds”, EMS Tracts Math., Vol. 13, Zurich, 2010. | MR 2683385 | Zbl 1244.57003

[8] E. M. Brown, Unknotted solid tori and genus one Whitehead manifolds, Trans. Amer. Math. Soc. 333 (1992), 835–847. | MR 1120774 | Zbl 0770.57007

[9] D. Burghelea, R. Lashof and M. Rothenberg, “Groups of Automorphisms of Manifolds”, Lecture Notes in Mathematics, Vol. 473, Springer-Verlag, 1975. | MR 380841 | Zbl 0307.57013

[10] J. Cerf, “Sur les difféomorphismes de la sphère de dimension trois (Γ 4 =0)”, Lecture Notes in Mathematics, Vol. 53, Springer, Berlin, 1968. | MR 229250 | Zbl 0164.24502

[11] P. T. Church and K. Lamotke, Non-trivial polynomial isolated singularities, Indag. Math. 37 (1975), 149–154. | MR 365592 | Zbl 0342.57017

[12] P. T. Church and J. G. Timourian, Differentiable maps with 0-dimensional critical set I, Pacific J. Math. 41 (1972), 615–630. | MR 315732 | Zbl 0236.58002

[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 336748 | Zbl 0279.57003

[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 350756 | Zbl 0281.57018

[15] P. E. Conner, On the impossibility of fibring a certain manifold by a compact fibre, Michigan Math. J. 5 (1957), 249–255. | MR 97063 | Zbl 0093.37304

[16] R. Daverman, “Decompositions of Manifolds”, Academic Press, 1986. | MR 872468 | Zbl 0608.57002

[17] A. Dimca, “Singularities and Topology of Hypersurfaces”, Springer-Verlag, Berlin, 1992. | MR 1194180 | Zbl 0753.57001

[18] A. Dold and H. Whitney, Classification of oriented bundles over a 4-complex, Ann. of Math. 69 (1959), 667–677. | MR 123331 | Zbl 0124.38103

[19] C. Earle and J. Eells Jr., A fibre bundle description of Teichmüller theory, J. Differential Geom. 3 (1969), 19–43. | MR 276999 | Zbl 0185.32901

[20] C. Earle and J. H. Schatz, Teichmüller theory for surfaces with boundary, J. Differential Geom. 4 (1970), 169–185. | MR 277000 | Zbl 0194.52802

[21] E. R. Fadell and S. Y. Husseini, “Geometry and Topology of Configuration Spaces”, Springer Monograph Math., 2001. | MR 1802644 | Zbl 0962.55001

[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 2550201 | Zbl 1192.57014

[23] A. Haefliger, Differentiable imbeddings, Bull. Amer. Math. Soc. 67 (1961), 109–112. | MR 123337 | Zbl 0099.18101

[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. | Numdam | MR 1611531 | Zbl 0127.13701

[25] A. Haefliger, Plongements différentiables dans le domaine stable, Comment. Math. Helv. 37 (1962/1963), 155–176. | MR 157391 | Zbl 0186.27302

[26] A. Haefliger, Differentiable embeddings of S n in S n+q for q>2, Ann. of Math. 83 (1966), 402–436. | MR 202151 | Zbl 0151.32502

[27] M.-E. Hamstrom, Regular mappings and the space of homeomorphisms on a 3-manifold, Mem. Amer. Math. Soc. 40 (1961), 42. | MR 152999 | Zbl 0116.39905

[28] A. Hatcher, Homeomorphisms of sufficiently large P 2 -irreducible 3-manifolds, Topology 15 (1976), 343–347. | MR 420620 | Zbl 0335.57004

[29] A. Hatcher, A proof of a Smale conjecture, Diff (S 3 )O(4), Ann. of Math. (2) 117 (1983), 553–607. | MR 701256 | Zbl 0531.57028

[30] A. Hatcher and D. McCullough, Finiteness of classifying spaces of relative diffeomorphism groups of 3-manifolds, Geom. Topol. 1 (1997), 91–109. | MR 1486644 | Zbl 0885.57008

[31] J. Hempel, “3-manifolds”, reprint of the 1976 original, AMS Chelsea Publishing, 2004. | MR 2098385 | Zbl 1058.57001

[32] W. Huebsch and M. Morse, Schoenflies extensions without interior differential singularities, Ann. of Math. 76 (1962), 18–54. | MR 146847 | Zbl 0111.35602

[33] W. Jaco, Three manifolds with fundamental group a free product, Bull. Amer. Math. Soc. 75 (1969), 972–977. | MR 243531 | Zbl 0184.48804

[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 68836 | Zbl 0067.15901

[35] L. Kauffman and W. Neumann, Products of knots, branched fibrations and sums of singularities, Topology 16 (1977), 369–393. | MR 488073 | Zbl 0447.57012

[36] H. C. King, Topological type of isolated singularities, Ann. of Math. 107 (1978), 385–397. | MR 494153 | Zbl 0354.57002

[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 1473242 | Zbl 0887.32012

[38] A. Kosinski, On the inertia group of π- manifolds, Amer. J. Math. 89 (1967), 227–248. | MR 214085 | Zbl 0172.25303

[39] A. Kosinski, “Differentiable Manifolds”, Academic Press, London, 1993. | Zbl 0767.57001

[40] E. Looijenga, A note on polynomial isolated singularities, Indag. Math. 33 (1971), 418–421. | MR 303557 | Zbl 0234.57010

[41] J. Milnor, On manifolds homeomorphic to the 7-sphere, Ann. of Math. (2) 64 (1956), 399–405. | MR 82103 | Zbl 0072.18402

[42] J. Milnor, Differentiable structures on spheres, Amer. J. Math. 81 (1959), 962–972. | MR 110107 | Zbl 0111.35501

[43] J. Milnor, A unique decomposition theorem for 3-manifolds, Amer. J. Math. 84 (1962), 1–7. | MR 142125 | Zbl 0108.36501

[44] J. Milnor, “Singular Points of Complex Hypersurfaces”, Princeton University Press, 1968. | MR 239612 | Zbl 0184.48405

[45] D. Montgomery and H. Samelson, Fiberings with singularities, Duke Math. J. 13 (1946), 51–56. | MR 15794 | Zbl 0060.41501

[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 1909641 | Zbl 1004.57027

[47] E. G. Rees, On a question of Milnor concerning singularities of maps, Proc. Edinburgh Math. Soc. (2) 43 (2000), 149–153. | MR 1744706 | Zbl 0974.58036

[48] D. Rolfsen, “Knots and Links”, corrected reprint of the 1976 original, Mathematics Lecture Series, 7, Publish or Perish, Inc., 1990. | MR 1277811 | Zbl 0854.57002

[49] L. Rudolph, Isolated critical points of mappings from R 4 to 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. | MR 920062 | Zbl 0626.57020

[50] R. Schultz, On the inertia group of a product of spheres, Trans. Amer. Math. Soc. 156 (1971), 137–153. | MR 275453 | Zbl 0216.45401

[51] P. Scott, There are no fake Seifert fibre spaces with infinite π 1 , Ann. of Math. (2) 117 (1983), 35–70. | MR 683801 | Zbl 0516.57006

[52] L. C. Siebenmann, Approximating cellular maps by homeomorphisms, Topology 11 (1972), 271–294. | MR 295365 | Zbl 0216.20101

[53] L. C. Siebenmann, Deformation of homeomorphisms of stratified sets, Comment. Math. Helv. 47 (1972), 123–163. | MR 319207 | Zbl 0252.57012

[54] E. H. Spanier, “Algebraic Topology”, Springer-Verlag, New York-Berlin, 1981. | MR 666554 | Zbl 0810.55001

[55] N. Steenrod, “The Topology of Fibre Bundles”, reprint of the 1957 edition, Princeton University Press, 1999. | MR 1688579 | Zbl 0942.55002

[56] F. Takens, Isolated critical points of C and C ω functions, Indag. Math. 29 (1967), 238–243. | MR 211419 | Zbl 0148.43501

[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. | MR 236942 | Zbl 0198.56603

[58] J. G. Timourian, Fiber bundles with discrete singular set, J. Math. Mech. 18 (1968), 61–70. | MR 235571 | Zbl 0169.25803

[59] F. Waldhausen, On irreducible 3-manifolds which are sufficiently large, Ann. of Math. (2) 87 (1968), 56–88. | MR 224099 | Zbl 0157.30603

[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 1243842 | Zbl 0841.57028

[61] J. H. C. Whitehead, On finite cocycles and the sphere theorem, Colloq. Math. 6(1958), 271–281. | MR 100843 | Zbl 0119.38605

[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 677482 | Zbl 0505.55017

[63] Yu Fengchun and Shicheng Wang, Covering degrees are determined by graph manifolds involved, Comment. Math. Helv. 74 (1999), 238–247. | MR 1691948 | Zbl 0930.57011