#### 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.

Notes on file formats
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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