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 . 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.

**[A]**P. Akhmetiev,*An elementary proof of Freedman's theorem on immersions*, St. Petersburg Math. J. 7 (1996), 749-754.**[B]**Thomas F. Banchoff,*Triple points and surgery of immersed surfaces*, Proc. Amer. Math. Soc.**46**(1974), 407–413. MR**0377897**, 10.1090/S0002-9939-1974-0377897-1**[F]**Michael H. Freedman,*Quadruple points of 3-manifolds in 𝑆⁴*, Comment. Math. Helv.**53**(1978), no. 3, 385–394. MR**505553**, 10.1007/BF02566085**[FT]**R. Fenn, P. Taylor,*On the number of triple points of an immersed surface*, (unpublished) preprint (1977).**[L]**Arunas Liulevicius,*A theorem in homological algebra and stable homotopy of projective spaces*, Trans. Amer. Math. Soc.**109**(1963), 540–552. MR**0156346**, 10.1090/S0002-9947-1963-0156346-X

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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

Email:
szucsandras@eudens.elte.hu

DOI:
https://doi.org/10.1090/S0002-9939-98-04083-0

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