Abstract
In the present work, we study the decompositions of codimension-one transitions that alter the singular set the of stable maps from \(S^3\) to \(\mathbb {R}^3,\) the topological behaviour of the singular set and the singularities in the branch set that involves cuspidal curves and swallowtails that alter the singular set. We also analyse the effects of these decompositions on the global invariants with prescribed branch sets.
Similar content being viewed by others
References
Arnold, V.I.: Topological Invariants of Plane Curves and Caustics. University Lecture Series 5. American Mathematical Society, Providence (1994)
Éliăsberg, J.: On singularities of folding type. Math. USSR-Izv. 4, 1119–1134 (1970)
Gibson, C.G.: Singular Points of Smooth Mappings. Reasearch Notes in Mathematics. Pitman, London (1978)
Goryunov, V.V.: Local invariants of maps between 3-manifolds. J. Topol. (2013). https://doi.org/10.1112/jtopol/jtt015
Hacon, D., Mendes de Jesus, C., Romero Fuster, MC.: Topological invariants of stable maps from a surface to the plane from a global viewpoint. In: Proceedings of the 6th Workshop on Real and Complex Singularities. Lecture Notes in Pure and Applied Mathematics, vol. 232, pp. 227–235. Marcel Dekker, New York (2003)
Hacon, D., de Jesus, Mendes C., Romero Fuster, M.C.: Stable maps from surfaces to the plane with prescribed branching data. Topol. Appl. 154, 166–175 (2007). https://doi.org/10.1016/j.topol.2006.04.005
Kálmán, T.: Stable maps of surfaces into the plane. Topol. Appl. 107(3), 307–316 (2000). https://doi.org/10.1016/S0166-8641(99)00105-4
Marar, W., Tari, F.: On the geometry of simple germs of co-rank 1 maps from \(\mathbb{R}^3\) to \(\mathbb{R}^3\). Math. Proc. Camb. Philos. Soc. 119(3), 469–481 (1996). https://doi.org/10.1017/S030500410007434X
Mather, J.N.: Stability of \(C^{\infty }\) Mappings VI: The Nice Dimensions. Proc. Liverpool Singularities-Sympos., I (1969/70), Lecture notes in Math., vol. 192, pp. 207–253. Springer, Berlin (1971). https://doi.org/10.1007/BFb0066809
Mendes de Jesus, C., Oset Sinha, R., Romero Fuster, M.C.: Global topological invariants of stable maps from 3-manifolds to \(R^{3}\). Proc. Steklov Inst. Math. 267, 205–216 (2009). https://doi.org/10.1134/S0081543809040178
Ohmoto, T., Aicardi, F.: First order local invariants of apparent contours. Topology 45, 27–45 (2006). https://doi.org/10.1016/j.top.2005.04.005
Oset Sinha, R.: Topological invariants of stable maps from 3-manifolds to three-space. PhD Dissertation, Valencia, p. 69 (2009)
Oset Sinha, R., Romero Fuster, M.C.: First-order local invariants of stable maps from 3-manifolds to \(\mathbb{R}^3\). Michigan Math. J 61, 385–414 (2012). https://doi.org/10.1307/mmj/1339011532
Oset Sinha, R., Romero Fuster, M.C.: Graphs of stable maps from 3-manifolds to 3-space. Mediterr. J. Math 10, 1107–1126 (2013). https://doi.org/10.1007/s00009-012-0224-2
Pignoni, R.: Projections of surfaces with a connected fold curve. Topol. Appl. 49, 55–74 (1993). https://doi.org/10.1016/0166-8641(93)90129-2
Saeki, O.: Simple stable maps of 3-manifolds into surfaces. Topology 35, 671–698 (1996). https://doi.org/10.1016/0040-9383(95)00034-8
Vassiliev, V.A.: Cohomology of knot spaces. Adv. Soviet. Math. 21, 23–69 (1990). https://doi.org/10.1090/advsov/001/03
Yamamoto, M.: First order semi-local invariants of stable maps of 3-manifolds into the plane. Proc. Lond. Math. Soc. (3) 92(2), 471–504 (2006). https://doi.org/10.1112/S0024611505015534
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
N. B. Huamaní was supported in part by Fondecyt C.G. 176-2015.
About this article
Cite this article
Huamaní, N.B., de Jesus, C.M. & Palacios, J. Invariants of Stable Maps from the 3-Sphere to the Euclidean 3-Space. Bull Braz Math Soc, New Series 50, 913–932 (2019). https://doi.org/10.1007/s00574-019-00133-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00574-019-00133-4