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A generalization of Banchoff's
triple point theorem

Authors: P. Akhmetiev, R. Rimányi and A. Szucs
Journal: Proc. Amer. Math. Soc. 126 (1998), 913-915
MSC (1991): Primary 57R42
MathSciNet review: 1422840
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Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [A] P. Akhmetiev, An elementary proof of Freedman's theorem on immersions, St. Petersburg Math. J. 7 (1996), 749-754.
  • [B] T. Banchoff, Triple points and surgery of immersed surfaces, Proc. Amer. Math. Soc. 46 (1974), 407-413. MR 51:14066
  • [F] M. Freedman, Quadruple points of 3-manifolds in $S^{4}$, Comment. Math. Helvetici 53 (1978), 385-394. MR 81b:57027
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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. Szucs
Affiliation: ELTE Department of Analysis, Budapest, Múzeum krt. 6–8, 1088, Hungary

Keywords: Immersion, multiple points
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
Article copyright: © Copyright 1998 American Mathematical Society

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