open access

https://doi.org/10.14943/84162

https://hdl.handle.net/2115/69821

Openings of differentiable map-germs and unfoldings

英語

2012-08-20

Department of Mathematics, Hokkaido University

Hokkaido University Preprint Series in Mathematics

1016

1-26

The algebraic notion of openings of a map-germ is introduced in this paper. An opening separates the self-intersections of the original mapgerm, preserving its singularities. The notion of openings is different from the notion of unfoldings. Openings do not unfold the singularities. For example, the swallowtail is an opening of the Whitney’s cusp mapgerm from plane to plane and the open swallowtail is a versal opening of them. Openings of map-germs appear as typical singularities in several problems of geometry and its applications. The notion of openings has close relations to isotropic map-germs in a symplectic space and integral map-germs in a contact space. We describe the openings of Morin singularities, namely, stable unfoldings of map-germs of corank one. The relation of unfoldings and openings are discussed. Moreover we provide a method to construct versal openings of map-germs and give versal openings of stable map-germs (R4, 0) ! (R4, 0). Lastly the relation of lowerable vector fields and openings is discussed.

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