The compression theorem

The compression theorem solves a 20 year old problem [2; problem 6]. Applications include short new 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. The proof introduces a novel technique in differential topology: proof by dynamical systems. We define flows which straighten vector fields and which then allow a given embedding or immersion to be ‘compressed’ to an immersion in a lower dimension. The technique gives explicit descriptions of the resulting immersions and can be seen as a way of desingularising certain maps. An example is the transition from the non-immersion of the projective plane in 3-space as a sphere with cross-cap to Boy’s surface. The result also gives a geometric meaning to the homotopy groups of the rack space [3, 4] and in particular we can construct a classifying space for codimension two embeddings with fundamental group mapping to a given group. It has also been applied by Wiest [17] to prove structure theorems for rack spaces. AMS Classification numbers Primary: 57R25, 57R27, 57R40, 57R42, 57R52 Secondary: 57R20, 57R45, 55P35, 55P40, 55P47