Manchester Geometry Seminar 2000/2001

Given a smooth map f: M -> Q how nice can the singularities of f be made by a small homotopy of f ? More generally suppose that M is embedded in Q x Rⁿ how nice can the singularities of the projection M -> Q be made by a small isotopy of M in Q x Rⁿ?

If "small" means small and smooth, then there are the classical Thom-Boardman singularities, which persist under small smooth isotopy. But if small means C⁰ small (which is what a topologist or a homotopy theorist would be interested in) then there is a great simplification in the singularities which persist, which then become closely related to the topology of the situation and are well understood in the metastable range.

References: Colin Rourke and Brian Sanderson,

• "The compression theorem"
  http://www.maths.warwick.ac.uk/~cpr/ftp/comp.ps or arxiv:math.GT/9712235
• "Directed embeddings: a short proof of Gromov's theorem"
  http://www.maths.warwick.ac.uk/~cpr/ftp/gromov.ps or arxiv:math.GT/0003026