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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Computing the Euler characteristic of a manifold with boundary


Author: Harold Levine
Journal: Proc. Amer. Math. Soc. 123 (1995), 2563-2567
MSC: Primary 57R45; Secondary 57R20, 58C25, 58C27
MathSciNet review: 1307546
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Abstract | References | Similar Articles | Additional Information

Abstract: Given a stable map, f, from an even-dimensional, compact manifold with boundary, M, into $ {\mathbb{R}^2}$, the Euler characteristic of $ M,\chi (M)$, is expressed in terms of projectivized rotation numbers of f restricted to $ S(f)$, the singular curves of f, and of f restricted to $ S(f\vert\partial M)$, the singular curves of f restricted to the boundary of M.


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

  • 1. H. Levine, A. O. Petters, and J. Wambsganss, Applications of singularity theory to gravitational lensing I; Globally stable systems (in preparation).
  • [1] Harold I. Levine, Mappings of manifolds into the plane, Amer. J. Math. 88 (1966), 357–365. MR 0208609 (34 #8418)
  • [2] M. Golubitsky and V. Guillemin, Stable mappings and their singularities, Springer-Verlag, New York-Heidelberg, 1973. Graduate Texts in Mathematics, Vol. 14. MR 0341518 (49 #6269)
  • [3] R. Thom, Les singularités des applications différentiables, Ann. Inst. Fourier, Grenoble 6 (1955–1956), 43–87 (French). MR 0087149 (19,310a)

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1995-1307546-0
PII: S 0002-9939(1995)1307546-0
Keywords: Stable map, singular curve, fold point, cusp point, rotation number, projectivized rotation number, Euler characteristic, manifold with boundary
Article copyright: © Copyright 1995 American Mathematical Society