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The compression theorem II: directed embeddings. (English) Zbl 1032.57028

This is the second of a set of three papers on the compression theorem. It contains a geometric proof of the normal deformation theorem, which is a general version of the compression theorem and was proved in the first paper of the series [Geom. Topol. 5, 399-429 (2001; Zbl 1002.57057)] by different methods. Central step in the present proof is a flattening lemma. It states that a compressible manifold \(M^n \subset Q^q \times {\mathbb{R}}^t\), \( q - n \geq 1\), may be deformed by a \(C^0\)-small isotopy in Q in a position, such that its tangent space lies in a given neighborhood of the horizontal subset of the Grassmann bundle of n-planes in the tangent bundle \(T(Q^q \times {\mathbb{R}}^t)\). Gromov’s theorem on directed embeddings [M. Gromov, Partial differential relations (Springer-Verlag, Berlin), (1986; Zbl 0651.53001)] is deduced from the normal deformation theorem. It is remarked that the normal deformation theorem may be deduced from Gromov’s theorem.

MSC:

57R25 Vector fields, frame fields in differential topology
57R40 Embeddings in differential topology
57R42 Immersions in differential topology
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References:

[1] M Gromov, Partial differential relations, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) 9, Springer (1986) · Zbl 0651.53001
[2] N H Kuiper, On \(C^1\)-isometric imbeddings I, II, Nederl. Akad. Wetensch. Proc. Ser. A. 58 (1955) 545, 683 · Zbl 0067.39601
[3] C Rourke, B Sanderson, The compression theorem I, Geom. Topol. 5 (2001) 399 · Zbl 1002.57057 · doi:10.2140/gt.2001.5.399
[4] C Rourke, B Sanderson, The compression theorem III: Applications, Algebr. Geom. Topol. 3 (2003) 857 · Zbl 1032.57029 · doi:10.2140/agt.2003.3.857
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