TOPOLOGICAL INVARIANTS AND MILNOR FIBER FOR A-FINITE GERMS C^2 → 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 → S^5 associated to finitely determined map germs (C^2 , 0) → (C^3 , 0).
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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
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