Namespaces
Variants
Actions

General position

From Encyclopedia of Mathematics
Jump to: navigation, search

generic position

A term used in such phrases as "the objects O in general (generic) position have the property S (or properties Si)" , "S is a property of general position" , "reduction (transformation) to general position" , the precise meaning of which depends on the context. Usually the set $\mathfrak{O}$ of all objects considered has a structure allowing certain subsets $\mathfrak{N} \subset \mathfrak{O}$ to be regarded as "small" , "negligible" or, conversely, "large" , "massive" ; $S$ is then regarded as a "property of general position" if the objects possessing it form a "large" subset of $\mathfrak{O}$. $\mathfrak{O}$ usually has one of the following structures: a) an algebraic variety; b) a differentiable manifold (possibly infinite-dimensional); c) a topological space, most often a Baire space in the first meaning of this term; or d) a measure space. The following are considered "small" respectively: algebraic subvarieties (of lower dimension), differentiable submanifolds and finite or countable unions of them, nowhere-dense sets or sets of the first category, sets of measure zero. A set $\mathfrak{A} \subset \mathfrak{O}$ is regarded as "large" if its complement is "small" . One then also says that $\mathfrak{A}$ contains "most" or "almost all" of the objects of $\mathfrak{O}$, and a property $S$ satisfied by almost all objects is called "typical" , or a property of general position. One frequently speaks of a "typical" object, or an object being of general position or an object in general position, implying (sometimes tacitly) that there are one or more "typical" properties (of what kind should be clear from the context) and that one is concerned with an object having these properties.

In a weaker sense, a "large" subset in cases c) and d) may mean a subset of the second category in a non-empty open subset of the space $\mathfrak{O}$ or a subset of positive measure. Then one says that this set of objects "cannot be neglected" (but one no longer says that it is "typical" ).

In cases a) and b) a "small" set $\mathfrak{A} \subset \mathfrak{O}$ has positive codimension $\operatorname{codim} \mathfrak{A}$. It is natural to say that as $\operatorname{codim} \mathfrak{A}$ gets larger, $\mathfrak{A}$ gets smaller. A situation close to b) (but more general) is when one can speak of $n$-parameter families of objects $O \left({\lambda_1, \dots, \lambda_n}\right)$, depending sufficiently smoothly on $n$ (scalar) parameters, and when all possible such families form a Baire space. If almost all (in sense c)) of these families do not contain objects of $\mathfrak{A}$, then one says that $\operatorname{codim} \mathfrak{A} > n$, and if this is true for any $n$, then one puts $\operatorname{codim} \mathfrak{A} = \infty$. Considerations of codimension play an important part in bifurcation theory and the theory of singularities of differentiable mappings (see also [8]).

Certain operations $g$ may act on the objects; the set $G$ of these operations is usually a group, or at least a semi-group with an identity $e$. One speaks of "reduction of an object O to general position by the operation g" when it is clear from the context what properties are being considered; "reduction" means that $gO$ has these properties. Like $\mathfrak{O}$, $G$ is usually provided with a structure allowing one to speak of a "large" set of operations, or to say that the operation $g$ taking $O$ to $gO$, with the necessary properties, can be chosen arbitrary close to $e$ ( "reduction to general position by a small shift" ).

For example, a line and a circle in the plane in general position either do not intersect or intersect in two points. In this case the object is the pair $\left({a, b}\right)$, where $a$ is the line and $b$ the circle, and the operations may be taken as Euclidean motions (or just parallel translations), acting on $a$ with $b$ fixed. The set of all possible objects $\mathfrak{O}$ and the group $G$ are naturally endowed with the structures mentioned above, and general position can be interpreted according to each of the versions. Originally, the term general position was used in similar geometrical problems, and thence it was transferred to areas of mathematics of a geometrical character, or at least considerably influenced by geometry (although arguments involving sets of the second category or of full measure are used outside these areas). To the present day, the term "general position" is often applied to a situation immediately generalizing the above example, when one speaks of the transversality of two submanifolds in some ambient manifold (or the related situation of transversality of the self-intersections of an immersed submanifold). In particular, in geometric topology (considering piecewise-linear or topological manifolds and the corresponding classes of mappings) the term "general position" is used almost exclusively as a synonym of transversality.

In algebraic geometry simple examples (like the one above) can easily be analyzed by elimination theory, where the ground field may be completely arbitrary (usually algebraically closed). There are theorems on general position in more complicated situations (for example, Bertini theorems, and the Lefschetz theorem on a hyperplane section); in studying the action of algebraic groups on algebraic varieties points in general position (cf. Point in general position) play a large part [1].

In differential topology and the theory of singularities of differentiable mappings, general position is used very extensively. Proofs are usually carried out by means of Sard's theorem or its corollaries, and the Abraham and Thom transversality theorems (see [2], [3]), which are more suitable for immediate application. In the infinite-dimensional case Sard's theorem is not valid, but weaker results that are sometimes sufficient can be obtained (see [4], [5], [14]).

There are several results on "typical" properties in the theory of smooth dynamical systems. Most of them are proved (particularly, in bifurcation theory) by reduction to Sard's theorem; positive results not connected with this reduction are few (see [6], [7], [9], [10], and the references to Rough system). A characteristic of the theory of smooth dynamical systems is the presence of an essential distinction between general position in the topological and metric sense (c) and d) above) , [15].

For general position in the differential geometry of manifolds see [12], [13].

References

[1] D. Mumford, "Geometric invariant theory" , Springer (1965) MR0214602 Zbl 0147.39304
[2] S. Lang, "Introduction to differentiable manifolds" , Interscience (1967) pp. App. III MR1931083 MR1532744 MR0155257 Zbl 1008.57001 Zbl 0103.15101
[3] V.I. Arnol'd, "Chapitres supplémentaires de la théorie des équations différentielles ordinaires" , MIR (1980) (Translated from Russian) Zbl 0956.34501
[4] J. Eells, J. McAlpin, "An approximate Morse–Sard theorem" J. Math. Mech. , 17 : 11 (1968) pp. 1055–1064 MR0226680 Zbl 0159.24901
[5] F. Quinn, "Transversal approximation on Banach manifolds" S.-S. Chern (ed.) S. Smale (ed.) , Global analysis , Proc. Symp. Pure Math. , 14 , Amer. Math. Soc. (1970) pp. 213–222 MR0264713 Zbl 0206.25705
[6] C. Gutierrez, "Structural stability of flows on the torus with a cross-cap" Trans. Amer. Math. Soc. , 241 (1978) pp. 311–320
[7] C. Gutierrez, "Smooth nonorientable nontrivial recurrence on two-manifolds" J. Differential Equations , 29 : 3 (1978) pp. 388–395 MR0507486 Zbl 0413.58018
[8] H. Kurland, J. Robbin, "Infinite codimension and transversality" A. Manning (ed.) , Dynamical systems (Warwick, 1974) , Lect. notes in math. , 468 , Springer (1975) pp. 135–150 MR0649273 Zbl 0317.58011
[9] F. Takens, "Tolerance stability" A. Manning (ed.) , Dynamical systems (Warwick, 1974) , Lect. notes in math. , 468 , Springer (1975) pp. 293–304 MR0650298 Zbl 0321.54022
[10] V.A. Dobrynskii, A.N. Sharkovskii, "Typicalness of dynamical systems almost all paths of which are stable under permanently acting perturbations" Soviet Math. Dokl. , 14 : 4 (1973) pp. 997–1000 Dokl. Akad. Nauk SSSR , 211 : 2 (1973) pp. 273–276
[11a] V.I. Arnol'd, "Small denominators, I. Mapping the circle onto itself" Izv. Akad. Nauk SSSR Ser. Mat. , 25 : 1 (1961) pp. 21–86 (In Russian)
[11b] V.I. Arnol'd, "Correction to "Small denominators, I. Mapping the circle onto itself" " Izv. Akad. Nauk SSSR Ser. Mat. , 28 : 2 (1964) pp. 479–480 (In Russian) Zbl 0369.58011
[12] C.T.C. Wall, "Geometric properties of generic differentiable manifolds" , Geometry and topology , Lect. notes in math. , 597 , Springer (1977) pp. 707–774 MR0494233 Zbl 0361.58004
[13] W. Klingenberg, "Lectures on closed geodesics" , Springer (1978) MR0478069 Zbl 0397.58018
[14] V.I. Arnol'd, S.M. [S.M. Khusein-Zade] Gusein-Zade, A.N. Varchenko, "Singularities of differentiable maps" , 1 , Birkhäuser (1985) (Translated from Russian) MR777682 Zbl 0554.58001
[15] V.I. Arnol'd, V.S. Afraimovich, Yu.S. Ilyashenko, L.P. Shil'nikov, "Dynamical systems" , 5 , Springer (forthcoming) (Translated from Russian)


Comments

For Thom's transversality theorem see Singularities of differentiable mappings and [a1]. In finite dimensions it was generalized to jet extensions [a2]. A simplified proof of the transversality theorem, together with a proof of Sard's theorem (cf. also Sard theorem), can be found, e.g., in [a3].

Finally, in situations where one has several structures (like in $\mathbf{R}$: a topological structure and the Lebesgue measure), the different notions of general position lead to paradoxal examples, see [a4].

References

[a1] R. Thom, "Un lemma sur les applications différentiable" Bol. Soc. Math. Mexico , 1 : 2 (1956) pp. 59–71
[a2] J.M. Boardman, "Singularities of differentiable maps" Publ. Math. IHES , 33 (1967) pp. 383–419 MR0231390 Zbl 0165.56803
[a3] Th. Bröcker, "Differentiable germs and catastrophes" , Cambridge Univ. Press (1975) MR0494220 Zbl 0302.58006
[a4] J.C. Oxtoby, "Measure and category" , Springer (1971) (Translated from German) MR0393403 Zbl 0217.09201
How to Cite This Entry:
General position. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=General_position&oldid=40174
This article was adapted from an original article by D.V. Anosov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article