Geometry & Topology Monographs 2 (1999), Proceedings of the Kirbyfest, paper no. 19, pages 343-348.

Cusp equivalence between smooth embeddings of the 2-sphere in 4-space

Takao Matumoto

Abstract. If the fundamental group of the complement of a smooth embedding f: S^2 \subset R^4 is a cyclic group, the map can be deformed to the standard embedding by a generic one-parameter family with at most cusp singularities. If two smooth embeddings are connected by such a deformation, they will be called cusp equivalent. We will discuss the relation of three equivalences of smooth 2-knots S^2 \subset R^4; cusp equivalence, stable equivalence and weakly stable equivalence.

Keywords. Cusp, generic deformation of maps, smooth 2-knots, stable equivalence

AMS subject classification. Primary: 57Q45. Secondary: 57R45.

E-print: arXiv:math.GT/9911257

Submitted: 30 December 1998. (Revised: 12 April 1999.) Published: 20 November 1999.

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Takao Matumoto
Department of Mathematics, Faculty of Science, Hiroshima University
Higashi-Hiroshima 739-8526, Japan
Email: matumoto@math.sci.hiroshima-u.ac.jp

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