Examples of smooth maps with finitely many critical points in dimensions $(4,3)$, $(8,5)$ and $(16,9)$
HTML articles powered by AMS MathViewer
- by Louis Funar, Cornel Pintea and Ping Zhang PDF
- Proc. Amer. Math. Soc. 138 (2010), 355-365 Request permission
Abstract:
We consider manifolds $M^{2n}$ which admit smooth maps into a connected sum of $S^1\times S^n$ with only finitely many critical points, for $n\in \{2,4,8\}$, and compute the minimal number of critical points.References
- Dorin Andrica and Louis Funar, On smooth maps with finitely many critical points, J. London Math. Soc. (2) 69 (2004), no. 3, 783–800. MR 2050046, DOI 10.1112/S0024610704005320
- D. Andrica, L. Funar and E. Kudryavtseva, The minimal number of critical points of maps between closed manifolds, Russian Journal of Mathematical Physics, special issue for the conference celebrating the 60th birthday of Nicolae Teleman (Ancona and Porto Nuovo, September 2007), J.-P. Brasslet, A. Legrand, R. Longo, A. Mishchenko, editors, to appear.
- Peter L. Antonelli, Structure theory for Montgomery-Samelson fiberings between manifolds. I, II, Canad. J. Math. 21 (1969), 170–179; ibid. 21 (1969), 180–186. MR 0238320, DOI 10.4153/cjm-1969-017-1
- Peter L. Antonelli, Differentiable Montgomery-Samelson fiberings with finite singular sets, Canadian J. Math. 21 (1969), 1489–1495. MR 261624, DOI 10.4153/CJM-1969-163-9
- Alexandru Dimca, Singularities and topology of hypersurfaces, Universitext, Springer-Verlag, New York, 1992. MR 1194180, DOI 10.1007/978-1-4612-4404-2
- Robert E. Gompf and András I. Stipsicz, $4$-manifolds and Kirby calculus, Graduate Studies in Mathematics, vol. 20, American Mathematical Society, Providence, RI, 1999. MR 1707327, DOI 10.1090/gsm/020
- André Haefliger, Differential embeddings of $S^{n}$ in $S^{n+q}$ for $q>2$, Ann. of Math. (2) 83 (1966), 402–436. MR 202151, DOI 10.2307/1970475
- William Huebsch and Marston Morse, Schoenflies extensions without interior differential singularities, Ann. of Math. (2) 76 (1962), 18–54. MR 146847, DOI 10.2307/1970263
- I. M. James and J. H. C. Whitehead, The homotopy theory of sphere bundles over spheres. I, Proc. London Math. Soc. (3) 4 (1954), 196–218. MR 61838, DOI 10.1112/plms/s3-4.1.196
- I. M. James and J. H. C. Whitehead, The homotopy theory of sphere bundles over spheres. II, Proc. London Math. Soc. (3) 5 (1955), 148–166. MR 68836, DOI 10.1112/plms/s3-5.2.148
- François Laudenbach and Valentin Poénaru, A note on $4$-dimensional handlebodies, Bull. Soc. Math. France 100 (1972), 337–344. MR 317343
- Mamoru Mimura, Homotopy theory of Lie groups, Handbook of algebraic topology, North-Holland, Amsterdam, 1995, pp. 951–991. MR 1361904, DOI 10.1016/B978-044481779-2/50020-1
- Reinhard Schultz, On the inertia group of a product of spheres, Trans. Amer. Math. Soc. 156 (1971), 137–153. MR 275453, DOI 10.1090/S0002-9947-1971-0275453-9
- J. G. Timourian, Fiber bundles with discrete singular set, J. Math. Mech. 18 (1968/1969), 61–70. MR 0235571, DOI 10.1512/iumj.1969.18.18007
Additional Information
- Louis Funar
- Affiliation: Institut Fourier BP 74, UMR 5582, Université de Grenoble I, 38402 Saint-Martin-d’Hères cedex, France
- Email: funar@fourier.ujf-grenoble.fr
- Cornel Pintea
- Affiliation: Department of Geometry, “Babeş-Bolyai” University, 400084 M. Kogălniceanu 1, Cluj-Napoca, Romania
- Email: cpintea@math.ubbcluj.ro
- Ping Zhang
- Affiliation: Department of Mathematics, Eastern Mediterranean University, Gazimag̃usa, North Cyprus, via Mersin 10, Turkey
- Email: ping.zhang@emu.edu.tr
- Received by editor(s): July 21, 2008
- Received by editor(s) in revised form: April 28, 2009
- Published electronically: September 3, 2009
- Communicated by: Paul Goerss
- © Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 138 (2010), 355-365
- MSC (2000): Primary 57R45, 55R55, 58K05, 57R60, 57R70
- DOI: https://doi.org/10.1090/S0002-9939-09-10028-X
- MathSciNet review: 2550201