The dual tree of a fold map germ from \(\mathbb{R}^3\) to \(\mathbb{R}^4 \). (English) Zbl 1518.58020
This paper explores the topological properties of analytic map germs \(f:(\mathbb{R}^3,0)\to(\mathbb{R}^4,0)\) with isolated instability. The associated link of this map corresponds to a stable map formed by the intersection between the image of f and a sufficiently small sphere \(S^3\) centered at the origin in \(\mathbb{R}^4\). In the case where f exhibits a fold type, the authors introduce a tree known as the dual tree, which contains all the topological information of the link. They demonstrate that this specific dual tree serves as a complete topological invariant, providing a comprehensive characterization of the fold-type map germs. Furthermore, the authors present a method to obtain normal forms for any topological class of fold type, offering practical applications of their results.
Reviewer: Erica Batista (Juazeiro do Norte)
MSC:
| 58K15 | Topological properties of mappings on manifolds |
| 58K40 | Classification; finite determinacy of map germs |
| 58K65 | Topological invariants on manifolds |
| 32S50 | Topological aspects of complex singularities: Lefschetz theorems, topological classification, invariants |
References:
| [1] | Batista, E. B., Costa, J. C. F. and Nuño Ballesteros, J. J.. The Reeb graph of a map germ from \(\mathbb{R}^3\) to \(\mathbb{R}^2\) with isolated zeros. Proc. Edinb. Math. Soc. 60 (2017), 319-348. · Zbl 1376.58016 |
| [2] | Batista, E. B., Costa, J. C. F. and Nuño Ballesteros, J. J.. The Reeb graph of a map germ from \(\mathbb{R}^3\) to \(\mathbb{R}^2\) with non isolated zeros. Bull. Braz. Math. Soc. (N.S.)49 (2018), 369-394. · Zbl 1415.58022 |
| [3] | Brown, M.. A proof of the generalised Schönflies theorem. Bull. Am. Math. Soc. 66 (1960), 74-76. · Zbl 0132.20002 |
| [4] | Casonatto, C., Romero Fuster, M. C. and Wik Atique, R.. Topological invariants of stable immersions of oriented 3-manifolds in \(\mathbb{R}^4 \). Topol. Appl. 159 (2012), 420-429. · Zbl 1255.57026 |
| [5] | Casonatto, C., Romero Fuster, M. C. and Wik Atique, R.. Topological invariants of stable maps of oriented 3-manifolds in \(\mathbb{R}^4 \). Geom. Dedicata194 (2018), 187-207. · Zbl 1395.57039 |
| [6] | Fukuda, T.. Local topological properties of differentiable mappings I. Invent. Math. 65 (1982), 227-250. · Zbl 0499.58008 |
| [7] | Houston, K. and Kirk, N.. On the classification and geometry of corank 1 map-germs from three-space to four-space. Singularity theory (Liverpool, 1996), xxii 325-351. London Mathematical Society Lecture Note Series 263 (Cambridge: Cambridge University Press, 1999). · Zbl 0945.58030 |
| [8] | Marar, W. L. and Nuño Ballesteros, J. J.. The doodle of a finitely determined map germ from \(\mathbb{R}^2\) to \(\mathbb{R}^3 \). Adv. Math. 221 (2009), 1281-1301. · Zbl 1171.58010 |
| [9] | Martins, R. and Nuño-Ballesteros, J. J.. Finitely determined singularities of ruled surfaces in \(\mathbb{R}^3 \). Math. Proc. Cambridge Philos. Soc. 147 (2009), 701-733. · Zbl 1183.53006 |
| [10] | Mather, J. N.. Stability of \(\mathcal{C}^{\infty } \)-mappings IV: classification of stable maps by \(\mathbb{R} \)-algebras. Inst. Hautes Études Sci. Publ. Math. 37 (1969), 223-248. · Zbl 0202.55102 |
| [11] | Mendes, R. and Nuño-Ballesteros, J. J.. Knots and the topology of singular surfaces in \(\mathbb{R}^4 \). Contemp. Math. 675 (2016), 229-239. · Zbl 1371.32005 |
| [12] | Mendes, R. and Nuño-Ballesteros, J. J.. Topological classification and finite determinacy of knotted maps. Michigan Math. J. 69 (2020), 831-848. · Zbl 1457.32015 |
| [13] | Mond, D.. On the classification of germs of maps from \(\mathbb{R}^2\) to \(\mathbb{R}^3 \). Proc. London Math. Soc. 50 (1985), 333-369. · Zbl 0557.58006 |
| [14] | Mond, D. and Nuño-Ballesteros, J. J.. Singularities of mappings. Grundlehren der Mathematischen Wissenschaften 357 (Springer, Cham, 2020). · Zbl 1448.58032 |
| [15] | Moya-Pérez, J. A. and Nuño-Ballesteros, J. J.. The link of a finitely determined map germ from \(\mathbb{R}^2\) to \(\mathbb{R}^2 \). J. Math. Soc. Jpn. 62 (2010), 1069-1092. · Zbl 1225.58018 |
| [16] | Moya-Pérez, J. A. and Nuño-Ballesteros, J. J.. Topological classification of corank 1 map germs from \(\mathbb{R}^3\) to \(\mathbb{R}^3 \). Rev. Mat. Complu. 27 (2014), 421-445. · Zbl 1315.58022 |
| [17] | Moya-Pérez, J. A. and Nuño-Ballesteros, J. J.. Gauss words and the topology of map germs from \(\mathbb{R}^3\) to \(\mathbb{R}^3 \). Rev. Mat. Iberoam. 31 (2015), 977-988. · Zbl 1335.58024 |
| [18] | Nuño-Ballesteros, J. J.. Combinatorial models in the topological classification of singularities of mappings. In Singularities and Foliations. Geometry and Topology and Applications, pp. 3-49. Springer Proceedings in Mathematics & Statistics 222 (Springer, Cham, 2018). · Zbl 1405.58021 |
| [19] | Wall, C. T. C.. Finite determinacy of smooth map germs. Bull. London Math. Soc. 13 (1981), 481-539. · Zbl 0451.58009 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
