Namespaces
Variants
Actions

Difference between revisions of "H-principle"

From Encyclopedia of Mathematics
Jump to: navigation, search
m (MR/ZBL numbers added)
m
 
(3 intermediate revisions by the same user not shown)
Line 1: Line 1:
 +
<!--This article has been texified automatically. Since there was no Nroff source code for this article,
 +
the semi-automatic procedure described at https://encyclopediaofmath.org/wiki/User:Maximilian_Janisch/latexlist
 +
was used.
 +
If the TeX and formula formatting is correct and if all png images have been replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category.
 +
 +
Out of 54 formulas, 53 were replaced by TEX code.-->
 +
 +
{{TEX|semi-auto}}{{TEX|part}}
 
''homotopy principle''
 
''homotopy principle''
  
 
A term having its origin in papers by M. Gromov in the 1960s and 1970s, some in collaboration with Y. Eliashberg and V. Rokhlin. It applies to partial differential equations or inequalities which have, very roughly speaking, as many solutions as predicted by topology.
 
A term having its origin in papers by M. Gromov in the 1960s and 1970s, some in collaboration with Y. Eliashberg and V. Rokhlin. It applies to partial differential equations or inequalities which have, very roughly speaking, as many solutions as predicted by topology.
  
The foundational example is the immersion theorem of S. Smale and M. Hirsch, which states the following: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120010/h1200102.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120010/h1200103.png" /> be smooth manifolds without boundary and suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120010/h1200104.png" /> or that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120010/h1200105.png" /> is non-compact. Then a smooth mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120010/h1200106.png" /> is homotopic (cf. also [[Homotopy|Homotopy]]) to a smooth immersion if and only if it can be covered by a continuous bundle mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120010/h1200107.png" /> which is injective on each fibre. The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120010/h1200108.png" />-principle provides a general language to formulate this and other geometric problems.
+
The foundational example is the immersion theorem of S. Smale and M. Hirsch, which states the following: Let $V$ and $W$ be smooth manifolds without boundary and suppose that $\dim V < \dim W$ or that $V$ is non-compact. Then a smooth mapping $f : V \rightarrow W$ is homotopic (cf. also [[Homotopy|Homotopy]]) to a smooth immersion if and only if it can be covered by a continuous bundle mapping $\varphi : T V \rightarrow T W$ which is injective on each fibre. The $h$-principle provides a general language to formulate this and other geometric problems.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120010/h1200109.png" /> be a smooth [[Fibration|fibration]] and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120010/h12001010.png" /> the space of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120010/h12001011.png" />-jets of smooth sections of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120010/h12001012.png" />. A section of the bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120010/h12001013.png" /> is called holonomic if it is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120010/h12001014.png" />-jet <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120010/h12001015.png" /> of a section <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120010/h12001016.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120010/h12001017.png" />. A differential relation (of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120010/h12001018.png" />) imposed on sections <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120010/h12001019.png" /> is a subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120010/h12001020.png" />. On says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120010/h12001021.png" /> satisfies the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120010/h12001023.png" />-principle if every section <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120010/h12001024.png" /> is homotopic to a holonomic section <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120010/h12001025.png" /> through a homotopy of sections <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120010/h12001026.png" />.
+
Let $\pi : X \rightarrow V$ be a smooth [[Fibration|fibration]] and let $X ^ { ( r ) }$ the space of $r$-jets of smooth sections of $\pi$. A section of the bundle $X ^ { ( r ) } \rightarrow V$ is called holonomic if it is the $r$-jet $J _ { f } ^ { r }$ of a section $f$ of $\pi$. A differential relation (of order $r$) imposed on sections $f : V \rightarrow X$ is a subset $\mathcal{R} \subset X ^ { ( r ) }$. One says that $\mathcal{R}$ satisfies the $h$-principle if every section $\sigma : V \rightarrow \mathcal{R}$ is homotopic to a holonomic section $J _ { f } ^ { r }$ through a homotopy of sections $V \rightarrow \mathcal{R}$.
  
There are several versions of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120010/h12001027.png" />-principle (relative, with parameters, etc.). For instance, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120010/h12001028.png" />-principle is called dense if a section of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120010/h12001030.png" /> can be homotoped into a holonomic section by a homotopy <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120010/h12001031.png" />-close to the original section.
+
There are several versions of the $h$-principle (relative, with parameters, etc.). For instance, the $h$-principle is called dense if a section of $\mathcal{R}$ can be homotoped into a holonomic section by a homotopy $C ^ { 0 }$-close to the original section.
  
To formulate the immersion theorem above in this language, one takes the trivial fibration <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120010/h12001032.png" /> and defines the immersion relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120010/h12001033.png" /> by stipulating that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120010/h12001034.png" /> consist of the injective linear mappings in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120010/h12001035.png" />. The immersion theorem then says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120010/h12001036.png" /> satisfies the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120010/h12001037.png" />-principle.
+
To formulate the immersion theorem above in this language, one takes the trivial fibration $X = V \times W \rightarrow V$ and defines the immersion relation $\mathcal{I} \subset X ^ { ( 1 ) }$ by stipulating that $\mathcal{I}_{ ( v , w )}$ consist of the injective linear mappings in $X _ { ( v , w ) } ^ { ( 1 ) } = \operatorname { Hom } ( T _ { v } V \rightarrow T _ { w } W )$. The immersion theorem then says that $\cal I$ satisfies the $h$-principle.
  
A surprising number of geometrically significant relations satisfy the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120010/h12001038.png" />-principle, and Gromov has developed powerful methods for proving the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120010/h12001039.png" />-principle. The fundamental reference for this subject is [[#References|[a1]]]. The principal methods for proving the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120010/h12001040.png" />-principle are removal of singularities, continuous sheaves, and [[Convex integration|convex integration]].
+
A surprising number of geometrically significant relations satisfy the $h$-principle, and Gromov has developed powerful methods for proving the $h$-principle. The fundamental reference for this subject is [[#References|[a1]]]. The principal methods for proving the $h$-principle are removal of singularities, continuous sheaves, and [[Convex integration|convex integration]].
  
The simplest instance of relations satisfying the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120010/h12001041.png" />-principle is arguably that of open relations over open manifolds (or in the case of extra dimension), subject to some naturality conditions. A very readable account of this theory can be found in [[#References|[a2]]]. One particular application of this form of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120010/h12001042.png" />-principle is to symplectic and contact geometry: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120010/h12001043.png" /> is open, then every non-degenerate <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120010/h12001044.png" />-form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120010/h12001045.png" /> (i.e. satisfying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120010/h12001046.png" />) is homotopic to an exact non-degenerate <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120010/h12001047.png" />-form, i.e. an exact symplectic form. Similarly, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120010/h12001048.png" /> is open and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120010/h12001049.png" /> are a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120010/h12001050.png" />-form, respectively a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120010/h12001051.png" />-form, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120010/h12001052.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120010/h12001053.png" /> is homotopic to a contact form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120010/h12001054.png" />, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120010/h12001055.png" />. Other important applications of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120010/h12001056.png" />-principle are the Nash–Kuiper <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120010/h12001058.png" />-isometric immersion theorem and the classification of isotropic immersions (see [[Ising model|Isotropic submanifold]]; [[Contact surgery|Contact surgery]]) in contact geometry.
+
The simplest instance of relations satisfying the $h$-principle is arguably that of open relations over open manifolds (or in the case of extra dimension), subject to some naturality conditions. A very readable account of this theory can be found in [[#References|[a2]]]. One particular application of this form of the $h$-principle is to symplectic and contact geometry: If $V ^ { 2 n }$ is open, then every non-degenerate $2$-form $\beta$ (i.e. satisfying $\beta ^ { n } \neq 0$) is homotopic to an exact non-degenerate $2$-form, i.e. an exact symplectic form. Similarly, if $V ^ { 2 n + 1 }$ is open and $\alpha , \beta$ are a $1$-form, respectively a $2$-form, with $\alpha \wedge \beta ^ { n } \neq 0$, then $\alpha$ is homotopic to a contact form $\gamma$, that is, $\gamma \wedge ( d \gamma ) ^ { n } \neq 0$. Other important applications of the $h$-principle are the Nash–Kuiper $C ^ { 1 }$-isometric immersion theorem and the classification of isotropic immersions (see [[Ising model|Isotropic submanifold]]; [[Contact surgery|Contact surgery]]) in contact geometry.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> M. Gromov, "Partial differential relations" , ''Ergebn. Math. Grenzgeb. (3)'' , '''9''' , Springer (1986) {{MR|0864505}} {{ZBL|0651.53001}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> A. Haefliger, "Lectures on the theorem of Gromov" , ''Proc. Liverpool Singularities Sympos. II'' , ''Lecture Notes Math.'' , '''209''' , Springer (1971) pp. 128–141 {{MR|0334241}} {{ZBL|0222.57020}} </TD></TR></table>
+
<table><tr><td valign="top">[a1]</td> <td valign="top"> M. Gromov, "Partial differential relations" , ''Ergebn. Math. Grenzgeb. (3)'' , '''9''' , Springer (1986) {{MR|0864505}} {{ZBL|0651.53001}} </td></tr><tr><td valign="top">[a2]</td> <td valign="top"> A. Haefliger, "Lectures on the theorem of Gromov" , ''Proc. Liverpool Singularities Sympos. II'' , ''Lecture Notes Math.'' , '''209''' , Springer (1971) pp. 128–141 {{MR|0334241}} {{ZBL|0222.57020}} </td></tr></table>

Latest revision as of 09:18, 4 July 2020

homotopy principle

A term having its origin in papers by M. Gromov in the 1960s and 1970s, some in collaboration with Y. Eliashberg and V. Rokhlin. It applies to partial differential equations or inequalities which have, very roughly speaking, as many solutions as predicted by topology.

The foundational example is the immersion theorem of S. Smale and M. Hirsch, which states the following: Let $V$ and $W$ be smooth manifolds without boundary and suppose that $\dim V < \dim W$ or that $V$ is non-compact. Then a smooth mapping $f : V \rightarrow W$ is homotopic (cf. also Homotopy) to a smooth immersion if and only if it can be covered by a continuous bundle mapping $\varphi : T V \rightarrow T W$ which is injective on each fibre. The $h$-principle provides a general language to formulate this and other geometric problems.

Let $\pi : X \rightarrow V$ be a smooth fibration and let $X ^ { ( r ) }$ the space of $r$-jets of smooth sections of $\pi$. A section of the bundle $X ^ { ( r ) } \rightarrow V$ is called holonomic if it is the $r$-jet $J _ { f } ^ { r }$ of a section $f$ of $\pi$. A differential relation (of order $r$) imposed on sections $f : V \rightarrow X$ is a subset $\mathcal{R} \subset X ^ { ( r ) }$. One says that $\mathcal{R}$ satisfies the $h$-principle if every section $\sigma : V \rightarrow \mathcal{R}$ is homotopic to a holonomic section $J _ { f } ^ { r }$ through a homotopy of sections $V \rightarrow \mathcal{R}$.

There are several versions of the $h$-principle (relative, with parameters, etc.). For instance, the $h$-principle is called dense if a section of $\mathcal{R}$ can be homotoped into a holonomic section by a homotopy $C ^ { 0 }$-close to the original section.

To formulate the immersion theorem above in this language, one takes the trivial fibration $X = V \times W \rightarrow V$ and defines the immersion relation $\mathcal{I} \subset X ^ { ( 1 ) }$ by stipulating that $\mathcal{I}_{ ( v , w )}$ consist of the injective linear mappings in $X _ { ( v , w ) } ^ { ( 1 ) } = \operatorname { Hom } ( T _ { v } V \rightarrow T _ { w } W )$. The immersion theorem then says that $\cal I$ satisfies the $h$-principle.

A surprising number of geometrically significant relations satisfy the $h$-principle, and Gromov has developed powerful methods for proving the $h$-principle. The fundamental reference for this subject is [a1]. The principal methods for proving the $h$-principle are removal of singularities, continuous sheaves, and convex integration.

The simplest instance of relations satisfying the $h$-principle is arguably that of open relations over open manifolds (or in the case of extra dimension), subject to some naturality conditions. A very readable account of this theory can be found in [a2]. One particular application of this form of the $h$-principle is to symplectic and contact geometry: If $V ^ { 2 n }$ is open, then every non-degenerate $2$-form $\beta$ (i.e. satisfying $\beta ^ { n } \neq 0$) is homotopic to an exact non-degenerate $2$-form, i.e. an exact symplectic form. Similarly, if $V ^ { 2 n + 1 }$ is open and $\alpha , \beta$ are a $1$-form, respectively a $2$-form, with $\alpha \wedge \beta ^ { n } \neq 0$, then $\alpha$ is homotopic to a contact form $\gamma$, that is, $\gamma \wedge ( d \gamma ) ^ { n } \neq 0$. Other important applications of the $h$-principle are the Nash–Kuiper $C ^ { 1 }$-isometric immersion theorem and the classification of isotropic immersions (see Isotropic submanifold; Contact surgery) in contact geometry.

References

[a1] M. Gromov, "Partial differential relations" , Ergebn. Math. Grenzgeb. (3) , 9 , Springer (1986) MR0864505 Zbl 0651.53001
[a2] A. Haefliger, "Lectures on the theorem of Gromov" , Proc. Liverpool Singularities Sympos. II , Lecture Notes Math. , 209 , Springer (1971) pp. 128–141 MR0334241 Zbl 0222.57020
How to Cite This Entry:
H-principle. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=H-principle&oldid=24469
This article was adapted from an original article by H. Geiges (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article