TOPOLOGICAL INVARIANTS AND MILNOR FIBRE FOR \(\mathcal{A}\)-FINITE GERMS \(C^2\) to \(C^3\)
DOI:
https://doi.org/10.37569/DalatUniversity.12.2.864(2022)Keywords:
Milnor fiber, Topological invariance.Abstract
This note is the observation that a simple combination of known results shows that the usual analytic invariants of a finitely determined multi-germ \(f : (C^2 , S) → (C^3 , 0) \)—namely, the image Milnor number , the number of cross-caps and triple points, \(C\) and \(T\), and the Milnor number \(μ(Σ)\) of the curve of double points in the target—depend only on the embedded topological type of the image of \(f\). As a consequence, one obtains the topological invariance of the sign-refined Smale invariant for immersions \(j : S^3 \looparrowright S^5\) associated to finitely determined map germs \((C^2 , 0) → (C^3 , 0)\).
Downloads
References
Buchweitz, R. O., & Greuel, G. M. (1980). The Milnor number and deformations of complex curve singularities. Inventiones Mathematicae, 58(3), 241-281. https://doi.org/10.1007/BF01390254
de Jong, T. (1990). The virtual number of d∞ points I. Topology, 29(2), 175-184. https://doi.org/10.1016/0040-9383(90)90006-6
Lê, D. T. (1973). Calcul du nombre de cycles évanouissants d’une hypersurface complexe [Calculation of the number of vanishing cycles of a complex hypersurface]. Annales de l’Institut Fourier, 23(4), 261-270. https://doi.org/10.5802/aif.491
Marar, W. L., & Mond, D. (1989). Multiple point schemes for corank 1 maps. Journal of the London Mathematical Society, 2(3), 553-567. https://doi.org/10.1112/jlms/s2-39.3.553
Marar, W. L., Nuño-Ballesteros, J. J., & Peñafort-Sanchis, G. (2012). Double point curves for corank 2 map germs from C2 to C3. Topology and its Applications, 159(2), 526-536. https://doi.org/10.1016/j.topol.2011.09.028
Massey, D., & Siersma, D. (1992). Deformation of polar methods. Annales de l’Institut Fourier, 42(4), 737-778. https://doi.org/10.5802/aif.1308
Mond, D. (1991). Vanishing cycles for analytic maps. In D. Mond & J. Montaldi (Eds.), Singularity theory and its applications, Warwick 1989, Part I: Geometric aspects of its singularities (pp. 221-234). Springer. https://doi.org/10.1007/BFb0086385
Mond, D., & Nuño-Ballesteros, J. J. (2020). Singularities of mappings: The local behaviour of smooth and complex analytic mappings. Springer. https://doi.org/10.1007/978-3-030-34440-5
Némethi, A., & Pintér, G. (2015). Immersions associated with holomorphic germs. Commentarii Mathematici Helvetici, 90(3), 513-541. https://doi.org/10.4171/CMH/363
Siersma, D. (1983). Isolated line singularities. In P. Orlik (Ed.), Singularities: Proceedings of symposia in pure mathematics (Vol. 40, Part 2, pp. 485-496). American Mathematical Society. https://doi.org/10.1090/pspum/040.2/713274
Siersma, D. (1988). Hypersurfaces with singular locus a plane curve and transversal type A1. Banach Center Publications, 20, 397-410. https://doi.org/10.4064/-20-1-397-410
Siersma, D. (1991). Vanishing cycles and special fibres. In D. Mond & J. Montaldi (Eds.), Singularity theory and applications, Warwick 1989, Part I: Geometric aspects of singularities (pp. 292-301). Springer. https://doi.org/10.1007/BFb0086389
Siersma, D. (2001). The vanishing topology of non isolated singularities. In D. Siersma, C. T. C. Wall, & V. Zakalyukin (Eds.), New developments in singularity theory, Nato science series (Vol. 21, pp. 447-472). Springer. https://doi.org/10.1007/978-94-010-0834-1_18
Siersma, D., & Tibăr, M. (2017). Milnor fibre homology via deformation. In W. Decker, G. Pfister, & M. Schulze (Eds.), Singularities and computer algebra: Festschrift for Gert-Martin Greuel on the occasion of his 70th birthday (pp. 305-322). Springer. https://doi.org/10.1007/978-3-319-28829-1_14
van Straten, D. (2017). On a theorem of Greuel and Steenbrink. In W. Decker, G. Pfister, & M. Schulze (Eds.), Singularities and computer algebra: Festschrift for Gert-Martin Greuel on the occasion of his 70th birthday (pp. 353-364). Springer. https://doi.org/10.1007/978-3-319-28829-1_17
Downloads
Published
Volume and Issues
Section
Copyright & License
Copyright (c) 2022 Javier Fernández de Bobadilla, Guillermo Peñafort-Sanchis, José Edson Sampaio.
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.
How to Cite
