Misplaced Pages

Pseudoisotopy theorem

Article snapshot taken from Wikipedia with creative commons attribution-sharealike license. Give it a read and then ask your questions in the chat. We can research this topic together.
On the connectivity of a group of diffeomorphisms of a manifold

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

Given a differentiable manifold M (with or without boundary), a pseudo-isotopy diffeomorphism of M is a diffeomorphism of M ×  which restricts to the identity on M × { 0 } ∪ ∂ M × [ 0 , 1 ] {\displaystyle M\times \{0\}\cup \partial M\times } .

Given f : M × [ 0 , 1 ] → M × [ 0 , 1 ] {\displaystyle f:M\times \to M\times } a pseudo-isotopy diffeomorphism, its restriction to M × { 1 } {\displaystyle M\times \{1\}} is a diffeomorphism g {\displaystyle g} 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 M × { t } {\displaystyle M\times \{t\}} for t ∈ [ 0 , 1 ] {\displaystyle t\in } .

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.

Relation to Cerf theory

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 π [ 0 , 1 ] ∘ f t {\displaystyle \pi _{}\circ f_{t}} . One then applies Cerf theory.

References

  1. ^ 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.
Categories: