Algebraic & Geometric Topology

The compression theorem III: applications

Colin Rourke and Brian Sanderson

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This is the third of three papers about the Compression Theorem: if Mm is embedded in Qq× with a normal vector field and if qm1, then the given vector field can be straightened (ie, made parallel to the given direction) by an isotopy of M and normal field in Q×.

The theorem can be deduced from Gromov’s theorem on directed embeddings [Partial differential relations, Springer–Verlag (1986); 2.4.5 C’] and the first two parts gave proofs. Here we are concerned with applications.

We give short new (and constructive) proofs for immersion theory and for the loops–suspension theorem of James et al and a new approach to classifying embeddings of manifolds in codimension one or more, which leads to theoretical solutions.

We also consider the general problem of controlling the singularities of a smooth projection up to C0–small isotopy and give a theoretical solution in the codimension 1 case.

Article information

Algebr. Geom. Topol., Volume 3, Number 2 (2003), 857-872.

Received: 31 January 2003
Revised: 16 September 2003
Accepted: 24 September 2003
First available in Project Euclid: 21 December 2017

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57R25: Vector fields, frame fields 57R27: Controllability of vector fields on C and real-analytic manifolds [See also 49Qxx, 37C10, 93B05] 57R40: Embeddings 57R42: Immersions 57R52: Isotopy
Secondary: 57R20: Characteristic classes and numbers 57R45: Singularities of differentiable mappings 55P35: Loop spaces 55P40: Suspensions 55P47: Infinite loop spaces

compression embedding isotopy immersion singularities vector field loops–suspension knot configuration space


Rourke, Colin; Sanderson, Brian. The compression theorem III: applications. Algebr. Geom. Topol. 3 (2003), no. 2, 857--872. doi:10.2140/agt.2003.3.857.

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