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The compression theorem I. (English) Zbl 1002.57057

Let \(M\) and \(Q\) be smooth manifolds of dimensions \(m\) and \(q\) respectively. The purpose of the present paper is to give a proof of the following Compression Theorem: Suppose that \(M\) is a compact manifold embedded in \(Q \times {\mathbf R}\) with a normal vector field and that \(q - m \geq 1\). Then \(M\) is isotopic to a compressible embedding (Theorem 2.1). Here \(M\) is called compressible if the vector field always points vertically up, namely, it points to the positive R-direction everywhere. This theorem implies that \(M\) can be isotoped to a position where it projects by R-directional projection to an immersion in \(Q\). The present paper is the first of a set of three papers about the comression theorem. Also the authors state as follows: “Here we give a direct proof that leads to an explicit description of the finishing embedding. In the second paper in the series we give a proof in the spirit of Gromov’s proof and in the third part we give applications.” The proof is certainly direct and constructive, but it needs a result of the appendix. The authors built up the basic results about the general position and transversality previous to the arguments in sections 2 and 4 and pack them into the appendix. The proof consists of the two steps. The first step makes the given normal field orthogonal to \(M\) and not downwards vertical everywhere (Corollary A.5). In the second step the resulting normal field is extended to a global unit vector field \(\gamma\) on \(Q \times {\mathbf R}\). This has the property that it has a positive vertical component everywhere and is a vertical up field in particular outside a tubular neighbourhood of \(M\) in \(Q \times {\mathbf R}\). Then it is easily seen that the given embedding \(M \subset Q \times {\mathbf R}\) can be isotoped to a compressible embedding by moving \(M\) along the flow defined by \(\gamma\). A great many geometrical fruits are hidden behind the proof given here. The authors take up the case where \(M\) has codimension 2 in \(Q \times {\mathbf R}\) and illustrate the construction of the isotopy given in the proof drawing a sequence of pictures which contains a considerable critical information. Such an observation yields for example the following extension of the theorem: Suppose that \(M\) is embedded in \(Q \times {\mathbf R}^n\) with \(n\) independent normal vector fields and that \(q - m \geq 1\). Then \(M\) is isotopic (by a \(C^0\)-small isotopy) to a parallel embedding (Multi-compression Theorem 4.5).

MSC:

57R25 Vector fields, frame fields in differential topology
57R27 Controllability of vector fields on \(C^\infty\) and real-analytic manifolds
57R40 Embeddings in differential topology
57R42 Immersions in differential topology
57R52 Isotopy in differential topology
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