A generalization of Banchoff’s triple point theorem
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- by P. Akhmetiev, R. Rimányi and A. Szűcs PDF
- Proc. Amer. Math. Soc. 126 (1998), 913-915 Request permission
Abstract:
Consider an immersion of a surface into $S^{3}$. Banchoff’s theorem states that the parity of the number of triple points and the parity of the Euler characteristic of the surface coincide. Here we generalize this theorem to codimension 1 immersions of arbitrary even dimensional manifolds in spheres. The proof is an analogue of a proof of Banchoff’s theorem circulated in preprint form due to R. Fenn and P. Taylor in 1977.References
- P. Akhmetiev, An elementary proof of Freedman’s theorem on immersions, St. Petersburg Math. J. 7 (1996), 749–754.
- Thomas F. Banchoff, Triple points and surgery of immersed surfaces, Proc. Amer. Math. Soc. 46 (1974), 407–413. MR 377897, DOI 10.1090/S0002-9939-1974-0377897-1
- Michael H. Freedman, Quadruple points of $3$-manifolds in $S^{4}$, Comment. Math. Helv. 53 (1978), no. 3, 385–394. MR 505553, DOI 10.1007/BF02566085
- R. Fenn, P. Taylor, On the number of triple points of an immersed surface, (unpublished) preprint (1977).
- Arunas Liulevicius, A theorem in homological algebra and stable homotopy of projective spaces, Trans. Amer. Math. Soc. 109 (1963), 540–552. MR 156346, DOI 10.1090/S0002-9947-1963-0156346-X
Additional Information
- P. Akhmetiev
- Affiliation: Institute of Terrestrial Magnetism and Radio Wave Propagation, Russian Academy of Sciences, Troitsk, Moscow Region 142092, Russia
- R. Rimányi
- Affiliation: ELTE Department of Geometry, Budapest, Rákóczi út 5., 1088, Hungary
- A. Szűcs
- Affiliation: ELTE Department of Analysis, Budapest, Múzeum krt. 6–8, 1088, Hungary
- Email: szucsandras@eudens.elte.hu
- Received by editor(s): July 4, 1995
- Received by editor(s) in revised form: September 2, 1996
- Additional Notes: The first author was supported in part by the Russian Foundation for Fundamental Research grant N 96-01-01166a. The second and the third authors were supported by the Hungarian National Science Foundation, Grant No. F-014906 and 4232, respectively
- Communicated by: Thomas Goodwillie
- © Copyright 1998 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 126 (1998), 913-915
- MSC (1991): Primary 57R42
- DOI: https://doi.org/10.1090/S0002-9939-98-04083-0
- MathSciNet review: 1422840