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.


57R25 Vector fields, frame fields in differential topology
57R40 Embeddings in differential topology
57R42 Immersions in differential topology
Full Text: DOI arXiv EuDML EMIS


[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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