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Milnor–Hamm Fibration for Mixed Maps

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Abstract

We consider a new class of singularities called mixed maps from Oka’s class. In this new setting we prove the existence of Milnor–Hamm fibration on the tube and sphere. Moreover, we discuss the problem of existence of a Milnor vector field for this class.

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Notes

  1. Usually the discriminant set \( {\text {Disc~}}G \) is just \( G({\text {Sing~}}G) \).

  2. This condition is also known in the literature as transversality property.

  3. It is also important to point out that under the conditions \({\text {Disc~}}G=\{0\}\) and (8) the map germ G is nice, see Massey (2010, Corollary 4.7).

  4. Two locally trivial smooth fibrations \( p:E \rightarrow B \) and \( p':E' \rightarrow B \) are said to be equivalent if there is a smooth diffeomorphism \( h:E\rightarrow E' \) such that \( p'\circ h = p \).

  5. For definition of MSL functions see Sect. 4 and for more details, see Araújo dos Santos and Ribeiro (2018) and Ribeiro (2019).

  6. For definitions and more details, see Oka (2018b, Section 2.3).

  7. For more details, see Gibson and Looijenga (1976) and Mather (2012).

  8. This result holds true in a more general setting where \( G:({\mathbb {R}}^m, 0) \rightarrow ({\mathbb {R}}^p,0) \) is an analytic map germ.

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Acknowledgements

The authors would like to thank the anonymous referee for his/her time reading our paper and for all valuable comments, suggestions and corrections. A. A. do Espírito Santo developed this paper during his post-doc period at Federal University of Alagoas. He would like to thank the Professor Juliana Roberta Theodoro de Lima and also his home-institution Federal University of Recôncavo da Bahia for supporting this period.

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Authors and Affiliations

  1. Departamento de Matemática, Universidade Federal do Espirito Santo, Vitoria, 29075-910, Brazil

    Maico F. Ribeiro

  2. Centro de Ciências Exatas e Tecnológicas, Universidade Federal do Recôncavo da Bahia, Cruz das Almas, 44380-000, Brazil

    Antonio A. do Espírito Santo

  3. Departamento de Matemática Aplicada, Universidade Federal do Espirito Santo, São Mateus, 29932-540, Brazil

    Fernando P. P. Reis

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  1. Maico F. Ribeiro
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  2. Antonio A. do Espírito Santo
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Correspondence to Maico F. Ribeiro.

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Ribeiro, M.F., do Espírito Santo, A.A. & Reis, F.P.P. Milnor–Hamm Fibration for Mixed Maps. Bull Braz Math Soc, New Series 52, 739–766 (2021). https://doi.org/10.1007/s00574-020-00229-2

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