Pseudoisotopy theorem

Summary

In mathematics, the pseudoisotopy theorem is a theorem of Jean Cerf's which refers to the connectivity of a group of diffeomorphisms of a manifold.

Statement edit

Given a differentiable manifold M (with or without boundary), a pseudo-isotopy diffeomorphism of M is a diffeomorphism of M × [0, 1] which restricts to the identity on  .

Given   a pseudo-isotopy diffeomorphism, its restriction to   is a diffeomorphism   of M. We say g is pseudo-isotopic to the identity. One should think of a pseudo-isotopy as something that is almost an isotopy—the obstruction to ƒ being an isotopy of g to the identity is whether or not ƒ preserves the level-sets   for  .

Cerf's theorem states that, provided M is simply-connected and dim(M) ≥ 5, the group of pseudo-isotopy diffeomorphisms of M is connected. Equivalently, a diffeomorphism of M is isotopic to the identity if and only if it is pseudo-isotopic to the identity.[1]

Relation to Cerf theory edit

The starting point of the proof is to think of the height function as a 1-parameter family of smooth functions on M by considering the function  . One then applies Cerf theory.[1]

References edit

  1. ^ a b Cerf, J. (1970). "La stratification naturelle des espaces de fonctions différentiables réelles et le théorème de la pseudo-isotopie". Inst. Hautes Études Sci. Publ. Math. 39: 5–173. doi:10.1007/BF02684687. S2CID 120787287.