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Algebraic Geometry and Singularities pp 259–276Cite as

How good are real pictures?

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Part of the book series: Progress in Mathematics ((PM,volume 134))

Abstract

This paper is motivated by some very good real pictures, drawn by Victor Goryunov in [10], which we reproduce here. Figure 1.1 shows images of stable perturbations of the A e -codimension 1 singularities of mappings ℂ2 → ℂ3. It is known (see [12], [17]) that the image of the complex mapping is homotopy equivalent to a 2-sphere: and Goryunov’s real pictures showed a real image with the same homotopy type. Goryunov was able to demonstrate, by elementary, though rather long, proofs (which are omitted in the published paper!), that in each case the inclusion of the real in the complex image, induced an isomorphism on the second homology group, so that his real pictures really did show the vanishing cycles associated with the codimension 1 singularities. The pictures in Figure 1.1 show stable perturbations of I: S 1 (birth of two cross-caps): II: Tangential contact of two immersed sheets; III: Cross-cap + immersed sheet; IV: Birth of two triple-points: and V: Quadruple point.

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Author information

Authors and Affiliations

  1. Mathematics Institute, Univ. of Warwick, Coventry, CV 4 7AL, UK

    David Mond

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  1. David Mond
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Editors and Affiliations

  1. Dep. de Algebra, Geometría y Topología Fac. de Ciencias, Universidad de Valladolid, Prado de la Magdalena s/n, 47005, Valladolid, Spain

    Antonio Campillo López

  2. Dept. of Algebra, Computatión, Geometría y Topología Fac. de Matemáticas, Universidad de Sevilla, Tarfia s/n, 41012, Sevilla, Spain

    Luis Narváez Macarro

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© 1996 Birkhäuser Verlag Basel/Switzerland

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Mond, D. (1996). How good are real pictures?. In: López, A.C., Macarro, L.N. (eds) Algebraic Geometry and Singularities. Progress in Mathematics, vol 134. Birkhäuser Basel. https://doi.org/10.1007/978-3-0348-9020-5_14

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  • DOI: https://doi.org/10.1007/978-3-0348-9020-5_14

  • Publisher Name: Birkhäuser Basel

  • Print ISBN: 978-3-0348-9870-6

  • Online ISBN: 978-3-0348-9020-5

  • eBook Packages: Springer Book Archive

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