Examples of Finitely Determined Map-Germs of Corank 3 Supporting Mond's μ ⩾ τ-Type Conjecture

ABSTRACT

In this paper, we give the first set of examples of finitely determined map-germs of corank 3 from 3-space to 4-space satisfying a conjecture by Mond that states the following. The number of parameters needed for a miniversal unfolding of a finitely determined map-germ from n-space to (n + 1)-space is less than (or equal to if the map-germ is weighted homogeneous) the rank of the nth homology group of the image of a stable perturbation of the map-germ, provided (n, n + 1) is in the range of Mather's nice dimensions. We describe some invariants of the multiple point spaces of one of our examples.

KEYWORDS:

• finite determinacy
• corank
• Mond conjecture
• image Milnor number

2000 AMS SUBJECT CLASSIFICATION:

• 58K40
• 32S30

Acknowledgments

The author would like to thank David Mond for his suggestions and especially for his comments on the proof of Proposition 3.1. She would also like to thank the referee for her/his constructive comments and bringing Ohmoto's formula for quadruple points to her attention.

Notes

1 Alternatively, see his preprint at arXiv:1309.0661v3, which may be more accessible to some readers.

2 The formula is a bit complicated and occupies a significantly large space. So, we will not repeat it here.

3 By a genuine k-tuple point, we refer to a point which has k distinct preimages under a stable perturbation.