## 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

### Keywords:

compression; embedding; isotopy; immersion; straightening; vector field
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### References:

 [1] J M Boardman, Singularities of differentiable maps, Inst. Hautes Études Sci. Publ. Math. (1967) 21 · Zbl 0165.56803 · doi:10.1007/BF02684585 [2] W Browder, W C Hsiang, Some problems on homotopy theory manifolds and transformation groups, Proc. Sympos. Pure Math., XXXII, Amer. Math. Soc. (1978) 251 · Zbl 0401.57002 [3] Y M Eliashberg, N M Mishachev, Wrinkling of smooth mappings II: Wrinkling of embeddings and K Igusa’s theorem, Topology 39 (2000) 711 · Zbl 0964.58028 · doi:10.1016/S0040-9383(99)00029-4 [4] R Fenn, C Rourke, B Sanderson, James bundles, Proc. London Math. Soc. $$(3)$$ 89 (2004) 217 · Zbl 1055.55005 · doi:10.1112/S0024611504014674 [5] M Gromov, Partial differential relations, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) 9, Springer (1986) · Zbl 0651.53001 [6] V Guillemin, A Pollack, Differential topology, Prentice-Hall (1974) · Zbl 0361.57001 [7] M W Hirsch, Immersions of manifolds, Trans. Amer. Math. Soc. 93 (1959) 242 · Zbl 0113.17202 · doi:10.2307/1993453 [8] I M James, Reduced product spaces, Ann. of Math. $$(2)$$ 62 (1955) 170 · Zbl 0064.41505 · doi:10.2307/2007107 [9] U Koschorke, B Sanderson, Geometric interpretations of the generalized Hopf invariant, Math. Scand. 41 (1977) 199 · Zbl 0385.55006 [10] J P May, The geometry of iterated loop spaces, Lecture Notes in Mathematics 271, Springer (1972) · Zbl 0244.55009 [11] J N Mather, Generic projections, Ann. of Math. $$(2)$$ 98 (1973) 226 · Zbl 0242.58001 · doi:10.2307/1970783 [12] R J Milgram, Iterated loop spaces, Ann. of Math. $$(2)$$ 84 (1966) 386 · Zbl 0145.19901 · doi:10.2307/1970453 [13] A Philips, Turning a surface inside out, Scientific American 214 (1966) 112 [14] C Rourke, B Sanderson, The compression theorem II: Directed embeddings, Geom. Topol. 5 (2001) 431 · Zbl 1032.57028 · doi:10.2140/gt.2001.5.431 [15] 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 [16] G Segal, Configuration-spaces and iterated loop-spaces, Invent. Math. 21 (1973) 213 · Zbl 0267.55020 · doi:10.1007/BF01390197 [17] S Smale, The classification of immersions of spheres in Euclidean spaces, Ann. of Math. $$(2)$$ 69 (1959) 327 · Zbl 0089.18201 · doi:10.2307/1970186 [18] B Wiest, Loop spaces and the compression theorem, Proc. Amer. Math. Soc. 128 (2000) 3741 · Zbl 0953.55007 · doi:10.1090/S0002-9939-00-05472-1 [19] B Wiest, Rack spaces and loop spaces, J. Knot Theory Ramifications 8 (1999) 99 · Zbl 0942.57006 · doi:10.1142/S0218216599000080
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