Abstract
Stable maps into the plane are good tools to obtain “views” of higher dimensional manifolds. We introduce the planar portraits to define the “view” properly. To start studying their relation to manifolds, we restrict our attention to their basic piece called the cusped fan. Fibreing structures over the cusped fan are studied and given a geometric characterisation. As by-products, we supply various stable maps and planar portraits of closed manifolds. In particular, two infiniteness properties of planar portraits are shown by using these examples.
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References
Blank, S.: Extending Immersions and Regular Homotopies in Codim 1. Thesis, Brandeis University (1967)
Buchstaber, V.M., Panov, T.E.: Torus Actions and Their Applications in Topology and Combinatorics. University Lecture Series, vol. 24. AMS, Providence (2002)
Crowley D., Escher C.M.: A classification of S 3-bundles over S 4. Differ. Geom. Appl. 18, 363–380 (2003)
Francis G., Troyer S.: Excellent maps with given folds and cusps. Houst. J. Math. 3, 165–194 (1977)
Fulton, W.: Introduction to toric varieties. Annals of Mathematics Studies, vol. 131. Princeton University Press, Princeton (1993)
Golubitsky, M., Guillemin, V.: Stable Mappings and Their Singularities. Graduate Texts in Mathematics, vol. 14. Springer, New York (1973)
Haefliger A.: Quelques remarques sur les applications différentiables d’une surface dans le plan. Ann. Inst. Fourier 10, 47–60 (1960)
Hacon D., Mendes de Jesus C., Romero Fuster M.C.: Fold maps from the sphere to the plane. Exp. Math. 15, 491–497 (2006)
Hacon D., Mendesde Jesus C., Romero Fuster M.C.: Stable maps from surfaces to the plane with prescribed branching data. Topol. Appl. 154, 166–175 (2007)
Hacon, D., Mendes de Jesus, C., Romero Fuster, M.C.: Topological invariants of stable maps from a surface to the plane from a global viewpoint. In: Real and Complex Singularities. Lecture Notes in Pure and Applied Mathematics, vol. 232, pp. 227–235. Dekker (2003)
Kobayashi M.: Simplifying certain stable mappings from simply connected 4-manifolds into the plane. Tokyo J. Math. 15, 327–349 (1992)
Levine H.: Elimination of cusps. Topology 3(suppl. 2), 263–296 (1965)
Levine H.: Mappings of manifolds into the plane. Am. J. Math. 88, 357–365 (1966)
Levine, H.: Classifying immersions into \({\mathbb{R}^4}\) over stable maps of 3-manifolds into \({\mathbb{R}^2}\) . Lecture Notes in Mathematics, vol. 1157. Springer, Berlin (1985)
Mather, J.: Stability of C ∞ mappings: VI. The nice dimensions. In: Proceedings of Liverpool Singularities Symposium I, Lecture Notes in Mathematics, vol. 192, pp. 207–253. Springer, Berlin (1971)
Motta W., Porto P., Saeki O.: Stable maps of 3-Manifolds into the plane and their quotient spaces. Proc. Lond. Math. Soc. 71, 158–174 (1995)
Pignoni R.: On surfaces and their contours. Manuscr. Math. 72, 223–249 (1991)
Steenrod N.E.: The Topology of Fibre Bundles. Princeton University Press, Princeton (1951)
Thom R.: Les singularités des applications différentiables. Ann. Inst. Fourier 6, 43–87 (1956)
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Kobayashi, M. On the cusped fan in a planar portrait of a manifold. Geom Dedicata 162, 25–43 (2013). https://doi.org/10.1007/s10711-012-9715-3
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DOI: https://doi.org/10.1007/s10711-012-9715-3