General position

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 of all objects considered has a structure allowing certain subsets to be regarded as "small" , "negligible" or, conversely, "large" , "massive" ; is then regarded as a "property of general position" if the objects possessing it form a "large" subset of . 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 is regarded as "large" if its complement is "small" . One then also says that contains "most" or "almost all" of the objects of , and a property 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 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 has positive codimension . It is natural to say that as gets larger, gets smaller. A situation close to b) (but more general) is when one can speak of -parameter families of objects , depending sufficiently smoothly on (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 , then one says that , and if this is true for any , then one puts . Considerations of codimension play an important part in bifurcation theory and the theory of singularities of differentiable mappings (see also [8]).

Certain operations may act on the objects; the set of these operations is usually a group, or at least a semi-group with an identity . 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 has these properties. Like , is usually provided with a structure allowing one to speak of a "large" set of operations, or to say that the operation taking to , with the necessary properties, can be chosen arbitrary close to ( "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 , where is the line and the circle, and the operations may be taken as Euclidean motions (or just parallel translations), acting on with fixed. The set of all possible objects and the group 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)
[2]	S. Lang, "Introduction to differentiable manifolds" , Interscience (1967) pp. App. III
[3]	V.I. Arnol'd, "Chapitres supplémentaires de la théorie des équations différentielles ordinaires" , MIR (1980) (Translated from Russian)
[4]	J. Eells, J. McAlpin, "An approximate Morse–Sard theorem" J. Math. Mech. , 17 : 11 (1968) pp. 1055–1064
[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
[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
[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
[9]	F. Takens, "Tolerance stability" A. Manning (ed.) , Dynamical systems (Warwick, 1974) , Lect. notes in math. , 468 , Springer (1975) pp. 293–304
[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)
[12]	C.T.C. Wall, "Geometric properties of generic differentiable manifolds" , Geometry and topology , Lect. notes in math. , 597 , Springer (1977) pp. 707–774
[13]	W. Klingenberg, "Lectures on closed geodesics" , Springer (1978)
[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)
[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 : 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
[a3]	Th. Bröcker, "Differentiable germs and catastrophes" , Cambridge Univ. Press (1975)
[a4]	J.C. Oxtoby, "Measure and category" , Springer (1971) (Translated from German)