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In [[mathematics]], the '''pseudoisotopy theorem''' is a theorem of Jean Cerf's<ref>[http://serge.mehl.free.fr/chrono/Cerf.html French mathematician, born 1928]</ref> 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'' × [0, 1] which restricts to the identity on <math>M \times \{0\} \cup \partial M \times [0,1]</math>. | |||
Given <math> f : M \times [0,1] \to M \times [0,1]</math> a pseudo-isotopy diffeomorphism, its restriction to <math>M \times \{1\}</math> is a diffeomorphism <math>g</math> of ''M''. We say ''g'' is ''pseudo-isotopic to the identity''. One should think of a pseudo-isotopy as something that is almost an [[homotopy|isotopy]]—the obstruction to ''ƒ'' being an isotopy of ''g'' to the identity is whether or not ''ƒ'' preserves the level-sets <math>M \times \{t\}</math> for <math> t \in [0,1]</math>. | |||
Cerf's theorem states that, provided ''M'' is [[Simply connected space|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. | |||
<ref>J.Cerf, La stratification naturelle des espaces de fonctions deff\'erentiables r\'eelles et le th\'eor\`eme de la pseudo-isotopie, Inst. Hautes \'Etudes Sci. Publ. Math. No {\bf 39} (1970) 5–173.</ref> | |||
== 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 <math>\pi_{[0,1]} \circ f_t</math>. One then applies [[Cerf theory]].<ref>J.Cerf, La stratification naturelle des espaces de fonctions différentiables réelles et le théorème de la pseudo-isotopie, Inst. Hautes Études Sci. Publ. Math. No. 39 (1970) 5–173.</ref> | |||
== References == | |||
<references/> | |||
[[Category:Theorems in differential topology]] | |||
[[Category:Singularity theory]] | |||
{{topology-stub}} | |||
Revision as of 20:57, 30 January 2014
In mathematics, the pseudoisotopy theorem is a theorem of Jean Cerf's[1] 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 × [0, 1] which restricts to the identity on .
![{\displaystyle M\times \{0\}\cup \partial M\times [0,1]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/099df1cfd31346e514d36ca72f5a135555a445cc)
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 .
![{\displaystyle f:M\times [0,1]\to M\times [0,1]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/779d9b26b13e03da2399c4cea78bd8f289a6744d)
![{\displaystyle t\in [0,1]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/31a5c18739ff04858eecc8fec2f53912c348e0e5)
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. [2]
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 . One then applies Cerf theory.[3]
![{\displaystyle \pi _{[0,1]}\circ f_{t}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4911c263f403072b2c138195c0746d345bd051a2)
References
- ↑ French mathematician, born 1928
- ↑ J.Cerf, La stratification naturelle des espaces de fonctions deff\'erentiables r\'eelles et le th\'eor\`eme de la pseudo-isotopie, Inst. Hautes \'Etudes Sci. Publ. Math. No {\bf 39} (1970) 5–173.
- ↑ J.Cerf, La stratification naturelle des espaces de fonctions différentiables réelles et le théorème de la pseudo-isotopie, Inst. Hautes Études Sci. Publ. Math. No. 39 (1970) 5–173.