Simplex moves on elementary surfaces
Working in the piecewise linear category throughout, the author considers surfaces in \(\mathbb{R}^4\): locally flat closed 2-dimensional submanifolds of \(\mathbb{R}^4\). A simplex move from one such surface \(F\) to another \(F'\) consists of a 3-simplex \(\sigma^3\) in \(\mathbb{R}^4\) such that \(F\cap\sigma^3 = F\cap\partial\sigma^3\) is a union of \(2\)-faces of \(\sigma^3\) and \(F'\) is the closure of \(F\cup\partial\sigma^3 \smallsetminus F\cap\partial\sigma^3\). Two surfaces \(F, F'\) in \(\mathbb{R}^4\) are ambient isotopic if and only if they are related by a finite sequence of simplex moves. The main result is about elementary surfaces: those for which the only critical points of the projection to the fourth \(\mathbb{R}\) factor are maxima, minima or saddle points. It is proved that two elementary surfaces are ambient isotopic if and only if after a small rotation about \(\mathbb{R}^2\) the two surfaces are related by a finite sequence of simplex moves through elementary surfaces. Crucial use is made of the \(\Lambda\)-moves of \textit{A. Kawauchi, T. Shibuya} and \textit{S. Suzuki} [Math. Semin. Notes, Kobe Univ. 10, 75-126 (1982; Zbl 0506.57014)].
scientific article; zbMATH DE number 1462598
scientific article