×

Smooth immersions of manifolds of low dimensions. (English. Russian original) Zbl 0842.57026

Russ. Acad. Sci., Sb., Math. 83, No. 1, 155-176 (1995); translation from Mat. Sb. 185, No. 10, 3-26 (1994).
Fix a Cartesian coordinate system \((x, y, z, t)\) in \(\mathbb{R}^4\) and the projection \(\Pi : \mathbb{R}^4 \to \mathbb{R}^3\), \(\Pi(x, y, z, t) = (x, y, t)\). Let \(\Psi : M^3 \to \mathbb{R}^4\) be a generic immersion of a smooth 3-dimensional manifold \(M^3\) into \(\mathbb{R}^4\).
The author studies possible positions of the stratified manifolds \(\Pi(\Psi(M^3)) \supset \Sigma^1 \supset \Sigma^{1,1} \supset \Sigma^{1,1,1}\) (consisting of the singular points of \(\Pi\)) and \(\Psi(M^3) \supset \Delta_2(\Psi) \supset \Delta_3(\Psi) \supset \Delta_4(\Psi)\) (consisting of the multiple points of the immersed manifold). To derive his main result, he decomposes \(\Psi\) into a \(t\)-parametrized one-parameter family of immersions \(\Psi_t : M^2_t \to \mathbb{R}^3_t(x,y,z)\), where, for any fixed \(\tau\), \(M^2_\tau = \Psi^{-1} (\Psi(M^3) \cap \mathbb{R}^3_\tau (x, y, z))\) and \(\mathbb{R}^3_\tau (x, y, z)\) is the hyperplane of the points in \(\mathbb{R}^4\) with the last coordinate \(t = \tau\).
Let \(\Pi_t : \mathbb{R}^3_t(x, y, z) \to \mathbb{R}^2_t(x,y)\) denote the \(\Pi\)-induced projection \(\Pi_t(x, y, z) = (x, y)\). Then the main result can be interpreted as a classification theorem for singularities of the mutual position of the curves \(\Pi_t(\Sigma^1)\) (consisting of the images of the critical points of \(\Pi_t\) restricted to \(\Psi_t(M^2_t))\) and \(\Pi_t(\Delta_2)\) (consisting of the selfintersection points of the immersed surface \(\Psi_t(M^2_t))\) in the plane.
The author gives several applications of his classification theorem, mainly to cobordism groups of immersions of surfaces into \(\mathbb{R}^3\) with oriented selfintersection curves.

MSC:

57R42 Immersions in differential topology
PDFBibTeX XMLCite
Full Text: DOI