View source for Singularities of differentiable mappings

A branch of mathematical analysis and differential geometry, in which those properties of mappings are studied which are preserved when the coordinates in the image and pre-image of the mapping are changed (or when changes are made which preserve certain supplementary structures); a general approach is proposed to the solution of various problems on degeneration of mappings, functions, vector fields, etc.; a classification is given of the most commonly encountered degenerations, and their normal forms, as well as algorithms which reduce to the normal forms, are determined.

A point of the domain of definition of a differentiable mapping (i.e. a mapping of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s0856201.png" />, see [[Differentiable manifold|Differentiable manifold]]) is said to be regular if the Jacobi matrix has maximum rank at this point, and critical in the opposite case.

The classical [[Implicit function|implicit function]] theorem describes the structure of a mapping in a neighbourhood of a regular point; in a neighbourhood of this point and in a neighbourhood of its image, there exist coordinates in which the mapping is linear.

In many cases it is not sufficient to confine the area of study simply to regular points; it is therefore natural to consider the following questions:

a) the description of a mapping in a neighbourhood of a critical point;

b) the description of the structure of the set of critical points.

For an arbitrary mapping there are no answers to a) and b), for two reasons: In attempting to deal with all mappings, there is no chance of obtaining explicit results (for example, the set of critical points can locally be an arbitrary closed set), and for practical applications it is sufficient to know the answers for only a large set of mappings.

The questions a) and b) and many others in the theory of singularities are studied along the following lines:

1) a set of  "untypical"  and  "pathological"  mappings is excluded from consideration;

2) a criterion of  "typicality"  of a mapping is determined;

3) it is ascertained that every mapping can be approximated by  "typical"  mappings;

4) the  "typical"  mappings are studied.

The choice of the set of typical mappings depends on the problem to be solved and is not unique: the fewer the mappings that are typical, the easier they are to study, although 2) and 3) require that the set of typical mappings is sufficiently broad and sufficiently constructively defined.

This scheme is illustrated by the Whitney theorem: Every differentiable mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s0856202.png" /> can be approximated by a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s0856203.png" /> such that for any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s0856204.png" /> in neighbourhoods of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s0856205.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s0856206.png" /> coordinates can be chosen at which the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s0856207.png" /> has one of the three normal forms:

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(for the typicality criterion, see [[#References|[3]]], [[#References|[4]]]). The work of H. Whitney (1955), in which this theorem was proved, is considered to be the beginning of the theory of singularities of differentiable mappings, although a number of individual results appeared far earlier ([[Morse theory|Morse theory]] of critical points of functions, Whitney's theorem on singularities of imbeddings, the work of L.S. Pontryagin on the connection between singularities and characteristic classes).

==Basic concepts in the theory of singularities of differentiable mappings.==


===Germs of differentiable mappings.===
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s0856209.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s08562010.png" /> be smooth manifolds, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s08562011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s08562012.png" />. (Hereafter, the term  "smooth"  will be used as a synonym for infinitely differentiable.) An equivalence class of mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s08562013.png" /> which coincide in a certain neighbourhood of the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s08562014.png" /> is called a germ at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s08562015.png" />; the set of germs of mappings which transfer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s08562016.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s08562017.png" /> is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s08562018.png" />. The group of germs of smooth changes of variables in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s08562019.png" /> which preserve the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s08562020.png" /> is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s08562021.png" />.

An important local problem in the theory of singularities of differentiable mappings is the study of the natural action of the group

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The solution of this and many similar problems is usually begun by approximating the function spaces and the infinite-dimensional groups acting on them by finite-dimensional manifolds and the actions on them by Lie groups. The results obtained are then transferred to the initial infinite-dimensional situation.

===Jet bundles.===
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s08562023.png" /> be smooth mappings, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s08562024.png" />; the mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s08562025.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s08562026.png" /> have, by definition, contact of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s08562028.png" /> at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s08562029.png" /> if their [[Taylor series|Taylor series]] at this point coincide up to the order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s08562030.png" />. An equivalence class of mappings which have contact of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s08562031.png" /> at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s08562032.png" /> is called a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s08562034.png" />-jet. The set of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s08562035.png" />-jets of mappings which transfer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s08562036.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s08562037.png" /> has the natural structure of a smooth manifold and is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s08562038.png" />. There is a well-defined natural projection

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s08562039.png" /></td> </tr></table>

The equivalence class of smooth changes of variables in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s08562040.png" /> which preserve the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s08562041.png" /> and have contact of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s08562042.png" /> at this point is called an invertible <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s08562044.png" />-jet at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s08562045.png" />. The invertible <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s08562046.png" />-jets form a Lie group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s08562047.png" />. The Lie group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s08562048.png" /> acts on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s08562049.png" /> and approximates the action of

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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s08562051.png" />. The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s08562052.png" /> has the natural structure of a smooth bundle over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s08562053.png" /> with fibre

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and structure group

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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s08562056.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s08562057.png" />.

===Singularities and classes of singularities.===
An orbit of the action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s08562058.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s08562059.png" /> is called a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s08562061.png" />-singularity; any subset in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s08562062.png" /> that is invariant under <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s08562063.png" /> is called a class of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s08562065.png" />-singularities. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s08562066.png" /> be such a class. Because <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s08562067.png" /> can be identified with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s08562068.png" />, the subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s08562069.png" /> is defined in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s08562070.png" />, irrespective of the method of identification. The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s08562071.png" /> is called a universal class of singularities (or a universal singularity if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s08562072.png" /> is a singularity). A universal singularity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s08562073.png" /> is a submanifold in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s08562074.png" /> of codimension equal to the codimension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s08562075.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s08562076.png" />.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s08562077.png" /> be a smooth mapping. The smooth mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s08562078.png" />, called the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s08562080.png" />-jet extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s08562081.png" />, is obtained by assigning to each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s08562082.png" /> the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s08562083.png" />-jet of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s08562084.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s08562085.png" />. The mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s08562086.png" /> has, by definition, a singularity of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s08562088.png" /> at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s08562089.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s08562090.png" />. The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s08562091.png" /> of all points at which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s08562092.png" /> has a singularity of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s08562093.png" /> is none other than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s08562094.png" />. The study of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s08562095.png" /> thus divides into two stages: the study of the universal set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s08562096.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s08562097.png" />, which reduces to the study of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s08562098.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s08562099.png" />, and the study of the mutual positions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620100.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620101.png" />. In the second stage one usually employs Thom's transversality theorem.

===Transversality.===
A smooth mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620102.png" /> of smooth manifolds is transversal to a submanifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620103.png" /> (denoted <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620104.png" />) if for any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620105.png" /> either <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620106.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620107.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620108.png" />, then the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620109.png" /> is either empty or is a submanifold in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620110.png" /> of codimension equal to the codimension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620111.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620112.png" />.

Thom's transversality theorem: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620113.png" /> be smooth manifolds and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620114.png" /> be a submanifold in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620115.png" />; then the set of those <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620116.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620117.png" /> is a massive subset in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620118.png" /> in the Whitney <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620119.png" />-topology. (A set is said to be massive (generic) if it is the intersection of a countable number of open dense subsets. A property is called generic if it holds for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620120.png" /> in a generic subset.)

The Whitney topology. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620121.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620122.png" /> be an open set in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620123.png" />. Let

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620124.png" /></td> </tr></table>

The sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620125.png" /> form the basis of some topology, called the Whitney <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620127.png" />-topology, on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620128.png" />. In this topology, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620129.png" /> is a [[Baire space|Baire space]], i.e. every massive subset is dense.

===Multi-jets.===
In the study of the self-intersection of the image of a smooth mapping, the concept of a multi-jet is used. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620130.png" /> be the natural projection. Let

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620131.png" /></td> </tr></table>

and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620132.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620133.png" /> times). The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620134.png" /> may be given the natural structure of a smooth manifold and is called an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620137.png" />-fold bundle of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620138.png" />-jets. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620139.png" />-fold jets, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620140.png" />-jet extension of a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620141.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620142.png" />-singularities, universal singularities, etc., are defined, and the analogue of Thom's transversality theorem has been proved.

===Stable differentiable mappings.===
A central problem in the early days of the theory of singularities of differentiable mappings was the study of stable differentiable mappings.

A smooth mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620143.png" /> of smooth manifolds is said to be stable if for any mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620144.png" /> that is sufficiently close to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620145.png" /> there exists diffeomorphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620146.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620147.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620148.png" />.

For small <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620149.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620150.png" />), as well as for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620151.png" /> and any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620152.png" />, stable differentiable mappings are dense in the space of all proper differentiable mappings [[#References|[3]]]. In the space of mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620153.png" />, stable mappings do not form an everywhere-dense set (see [[#References|[1]]]). For certain pairs of manifolds (for example, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620154.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620155.png" />) there is no stable mapping from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620156.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620157.png" />. All  "stable dimensions"  <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620158.png" /> have been found in [[#References|[14]]], : For any smooth manifolds <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620159.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620160.png" />, the stable mappings from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620161.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620162.png" /> are dense in the space of proper differentiable mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620163.png" />, equipped with the Whitney <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620164.png" />-topology, if and only if the pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620165.png" /> satisfies at least one of the following conditions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620166.png" />: a) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620167.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620168.png" />; b) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620169.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620170.png" />; c) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620171.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620172.png" />; d) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620173.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620174.png" />; e) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620175.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620176.png" />.

In proving this theorem, as well as in many other questions, the following two concepts prove to be useful: A mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620177.png" /> is called homotopy stable if for any smooth homotopy <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620178.png" /> of the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620179.png" /> there are smooth homotopies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620180.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620181.png" /> of the identity diffeomorphisms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620182.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620183.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620184.png" /> for sufficiently small <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620185.png" />. A mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620186.png" /> is called infinitesimal stable if every mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620187.png" /> that is infinitely close to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620188.png" /> can be obtained from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620189.png" /> by diffeomorphisms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620190.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620191.png" /> which are  "infinitely close to the identity" . For a proper mapping, the concepts of stability, homotopy stability and infinitesimal stability coincide [[#References|[3]]]. The problem of finding local normal forms of stable mappings reduces to the problem of classifying certain finite-dimensional local algebras [[#References|[14]]], . For fixed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620192.png" />, the number of such normal forms is finite.

If in the definition of a stable mapping, homeomorphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620193.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620194.png" /> are taken instead of diffeomorphisms, then the definition of a topologically stable mapping is obtained. The theorem on the density of the set of topological stable mappings in the set of all mappings of any compact manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620195.png" /> into any manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620196.png" /> (for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620197.png" />) has been proved (see [[#References|[8]]]).

===Finitely defined germs.===
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620198.png" /> be some equivalence relation on the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620199.png" /> of germs of mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620200.png" /> which take <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620201.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620202.png" />. The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620203.png" />-jet of any such germ is the segment of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620204.png" /> of its Taylor series. A germ <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620205.png" /> is said to be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620207.png" />-defined if any other germ <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620208.png" /> with the same <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620209.png" />-jet satisfies the relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620210.png" />. A germ is finitely defined if it is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620211.png" />-defined for a certain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620212.png" />. A <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620213.png" />-jet <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620214.png" /> such that any two germs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620215.png" /> which have <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620216.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620217.png" />-jet satisfy the relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620218.png" />, is said to be sufficient. The most commonly encountered equivalences have special names:

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620221.png" />-equivalence — belonging to one orbit of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620222.png" /> of  "correct"  changes of coordinates.

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620224.png" />-equivalence — belonging to one orbit of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620225.png" />.

Topological equivalence — belonging to one orbit of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620226.png" />.

The study of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620227.png" />-defined germs reduces to the study of mappings defined by polynomials of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620228.png" />.

The question of whether a germ is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620229.png" />-defined with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620230.png" />-equivalence reduces to the problem of the solvability of an explicit system of finitely many linear equations.

The set of finitely defined germs with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620231.png" />-equivalence is open in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620232.png" />, but is not dense for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620233.png" />. It is natural to consider the coarser relation of topological equivalence. After rejection of a subset of finite codimension from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620234.png" />, a countable number of topological equivalence classes remains, each of which is a semi-algebraic set. It follows that the mappings with germs that are topologically equivalent to polynomial ones form an open dense set in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620235.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620236.png" /> is compact) .

===Deformation.===
If a mapping depends on parameters, then a family of mappings is said to be defined. If the family of mappings is studied locally, then when the parameters are changed slightly in a neighbourhood of fixed values, one speaks of a deformation of the mapping corresponding to these fixed values of the parameters. In many cases, the study of all possible deformations reduces to the study of a single natural deformation, from which all others are obtained. This deformation, in a sense the largest, contains all essentially different deformations of the given mapping. It is called the versal deformation (see , [[#References|[12]]], ).

==Critical points of functions.==
A critical point of a function is called non-degenerate if the second differential is a non-degenerate quadratic form. A function in general position has only non-degenerate critical points, while in a neighbourhood of each of them it can be reduced to a standard form. Degenerate critical points are studied when these functions depend on parameters, and the greater the number of these parameters, the more complex critical points will be encountered that are non-removable (by a small shift) for certain values of the parameters.

A family of functions depending on any number of parameters can be transformed by a small shift into a family in which, for every value of the parameter in a neighbourhood of any point of the domain of definition, the function is represented by a polynomial in a local system of coordinates. This means that in the local study of functions it is possible to examine only polynomials and to use complex analysis.

===Classification.===
It is natural to begin by classifying the germs at 0 of holomorphic functions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620237.png" />, taking two germs to be equivalent if one can be transferred into the other by the germ of a holomorphic change of coordinates in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620239.png" /> which preserves 0. A jet (Taylor polynomial) of a holomorphic function at 0 is sufficient if it defines the function up to equivalence. A germ whose critical point 0 is isolated always has a sufficient jet and, consequently, is equivalent to a polynomial. The number of non-degenerate critical points into which the critical point 0 splits when the function is shifted slightly is called the multiplicity (or Milnor number) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620240.png" /> of the critical point 0. If the multiplicity of a critical point of a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620241.png" /> is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620242.png" />, then the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620243.png" />-jet is sufficient. Since the multiplicity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620244.png" /> cannot increase under a small change of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620245.png" />, the classification of functions close to a function with an isolated critical point reduces to the study of the action of the Lie group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620246.png" />-jets of changes of variables on the space of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620247.png" />-jets for a sufficiently large <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620248.png" />. In a space of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620249.png" />-jets of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620250.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620251.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620252.png" />, the codimension of the orbit of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620253.png" /> is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620254.png" />; thus, the critical points of multiplicity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620255.png" /> are non-removable in families of functions which depend on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620256.png" /> parameters. A classification (see [[#References|[10]]]) of all critical points of multiplicity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620257.png" /> and an algorithm for reducing any such function to a normal form have been obtained. The complexity of a critical point is defined not only by its multiplicity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620258.png" />, but also by its modality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620259.png" /> (the number of moduli). A critical point is simple (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620261.png" />-modal) if among the critical points near to it there are at most finitely many pairwise non-equivalent points. Two germs of functions are called stably equivalent if they become equivalent after being added directly to non-degenerate quadratic forms in the appropriate number of variables (for germs of functions in the same number of variables, stable equivalence is the same ordinary equivalence).

Simple germs are exhausted up to stable equivalence by the following list:

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620262.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620263.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620264.png" />;

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620265.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620266.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620267.png" />;

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620268.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620269.png" />;

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620270.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620271.png" />;

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620272.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620273.png" />.

The modality of a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620274.png" /> under the action of a Lie group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620275.png" /> on a manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620276.png" /> is the smallest number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620277.png" /> such that a sufficiently small neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620278.png" /> is covered by a finite number of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620279.png" />-parametric families of orbits.

A classification of germs of functions of modality 1 and 2 has also been obtained (see [[#References|[10]]]). The classification of simple singularities and singularities of small modality is related to Lie groups, the Coxeter and Weyl series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620280.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620281.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620282.png" />, the theory of Artin braids, the classification of regular polyhedra in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620283.png" />-dimensional space, the Kodaira classification of degenerate elliptic curves, and the classification of triangles in the Lobachevskii plane (see [[#References|[10]]], ).

===Boundary singularities.===
A series of geometric problems requires the study of critical points of functions on a manifold with boundary.

In the complex case this situation corresponds to the study of the germ of a function defined in the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620284.png" /> with a distinguished subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620285.png" />. These germs are studied up to changes of variables in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620286.png" /> which transfer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620287.png" /> onto itself. In this situation, a classification has also been obtained of all simple germs, and of germs of modalities 1 and 2. The classification of simple boundary singularities proves to be related to the simple Lie algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620288.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620289.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620290.png" />.

==Topological characteristics of the germ of a holomorphic function.==
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620291.png" /> be a holomorphic function in a neighbourhood of zero, with a critical point of multiplicity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620292.png" /> at zero. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620293.png" /> be positive numbers, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620294.png" /> be the ball <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620295.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620296.png" /> be its boundary, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620297.png" /> be the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620298.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620299.png" /> be the punctured disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620300.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620301.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620302.png" />. For appropriate <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620303.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620304.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620305.png" /> is sufficiently small and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620306.png" /> is sufficiently small in relation to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620307.png" />), the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620308.png" /> is a smooth locally trivial fibration. The fibre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620309.png" /> of this fibration is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620310.png" />-dimensional manifold with a boundary that is homotopically equivalent to a bouquet of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620311.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620312.png" />-dimensional spheres. The boundary of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620313.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620314.png" />-dimensional manifold which is diffeomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620315.png" />. Even for relatively simple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620316.png" />, this manifold can be non-trivial. For example, the 28 manifolds

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620317.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620318.png" /></td> </tr></table>

are 28 Milnor spheres (which are all homeomorphic to an ordinary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620319.png" />-dimensional sphere, but are not pairwise diffeomorphic). The reduced homology group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620320.png" /> is isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620321.png" />. The intersection index defines an integer-valued bilinear form on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620322.png" />. The transfer of fibres of the fibration <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620323.png" /> along curves in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620324.png" /> defines the action of the fundamental group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620325.png" /> in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620326.png" />-dimensional homology space of the fibre. The automorphism of the homology group corresponding to the generator of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620327.png" /> is called the monodromy operator. The monodromy operator retains the intersection form. The eigenvalues of the monodromy operator contain information on the asymptotics of various integrals related to the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620328.png" />.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  V.I. Arnol'd,   S.M. [S.M. Khusein-Zade] Gusein-Zade,   A.N. Varchenko,   "Singularities of differentiable maps" , '''1–2''' , Birkhäuser  (1985–1988)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  V.I. Arnol'd,   "Mathematical methods of classical mechanics" , Springer  (1978)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  M. Golubitskii,   V. Guillemin,   "Stable mappings and their singularities" , Springer  (1973)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  P. Bröcker,   L. Lander,   "Differentiable germs and catastrophes" , Cambridge Univ. Press  (1975)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  T. Poston,   I. Stewart,   "Catastrophe theory and its applications" , Pitman  (1978)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  J. Milnor,   "Singular points of complex hypersurfaces" , Princeton Univ. Press  (1968)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> , ''Singularities of differentiable mappings'' , Moscow  (1968)  (In Russian; translated from English and French)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  C.G. Gibson,   K. Wirthmüller,   A.A. du Plessis,   E.J.N. Looijenga,   "Topological stability of smooth mappings" , ''Lect. notes in math.'' , '''552''' , Springer  (1976)</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  R. Thom,   "Structural stability and morphogenesis" , Benjamin  (1966)  (Translated from French)</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  V.I. Arnol'd,   "Normal forms for functions near degenerate critical points, the Weyl groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620329.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620330.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620331.png" /> and Lagrangian singularities"  ''Funct. Anal. Appl.'' , '''6''' :  4  (1972)  pp. 254–272  ''Funkts. Anal. i Prilozhen.'' , '''6''' :  4  (1972)  pp. 3–25</TD></TR><TR><TD valign="top">[11a]</TD> <TD valign="top">  V.I. Arnol'd,   "Lectures on bifurcation in versal families"  ''Russian Math. Surveys'' , '''27''' :  5  (1972)  pp. 54–123  ''Uspekhi Mat. Nauk'' , '''27''' :  5  (1972)  pp. 119–184</TD></TR><TR><TD valign="top">[11b]</TD> <TD valign="top">  V.I. Arnol'd,   "Remarks on the stationary phase method and Coxeter numbers"  ''Russian Math. Surveys'' , '''28''' :  5  (1973)  pp. 19–48  ''Uspekhi Mat. Nauk'' , '''28''' :  5  (1973)  pp. 17–44</TD></TR><TR><TD valign="top">[11c]</TD> <TD valign="top">  V.I. Arnol'd,   "Normal forms of functions in a neighbourhood of a degenerate critical point"  ''Russian Math. Surveys'' , '''29''' :  2  (1974)  pp. 1–5  ''Uspekhi Mat. Nauk'' , '''29''' :  2  (1974)  pp. 11–49</TD></TR><TR><TD valign="top">[11d]</TD> <TD valign="top">  V.I. Arnol'd,   "Critical points of smooth functions and their normal forms"  ''Russian Math. Surveys'' , '''30''' :  5  (1975)  pp. 1–75  ''Uspekhi Mat. Nauk'' , '''30''' :  5  (1975)  pp. 3–65</TD></TR><TR><TD valign="top">[12]</TD> <TD valign="top">  R. Thom,   "The bifurcation subspace of a set of maps"  N.H. Kuiper (ed.) , ''Manifolds (Amsterdam, 1970)'' , ''Lect. notes in math.'' , '''197''' , Springer  (1971)  pp. 202–208</TD></TR><TR><TD valign="top">[13a]</TD> <TD valign="top">  A.N. Varchenko,   "Local topological properties of differentiable mappings"  ''Math. USSR Izv.'' , '''8''' :  5  (1974)  pp. 1033–1082  ''Izv. Akad. Nauk SSSR'' , '''38''' :  5  (1974)  pp. 1037–1090</TD></TR><TR><TD valign="top">[13b]</TD> <TD valign="top">  A.N. Varchenko,   "A theorem on topological versal deformations"  ''Math. USSR Izv.'' , '''9''' :  2  (1975)  pp. 277–296  ''Izv. Akad. Nauk SSSR'' , '''39''' :  2  (1975)  pp. 294–314</TD></TR><TR><TD valign="top">[14]</TD> <TD valign="top">  J. Mather,   "Stability of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620332.png" />-mappings III"  ''Publ. Math. IHES'' , '''35'''  (1969)  pp. 127–156; 279–308</TD></TR><TR><TD valign="top">[15a]</TD> <TD valign="top">  J. Mather,   "Stability of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620333.png" />-mappings IV"  ''Publ. Math. IHES'' , '''37'''  (1970)  pp. 223–248</TD></TR><TR><TD valign="top">[15b]</TD> <TD valign="top">  J. Mather,   "Stability of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620334.png" />-mappings V"  ''Adv. Math.'' , '''4'''  (1970)  pp. 301–335</TD></TR><TR><TD valign="top">[15c]</TD> <TD valign="top">  J. Mather,   "Stability of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085620/s085620335.png" />-mappings VI"  C.T.C. Wall (ed.) , ''Proc. Liverpool Singularities Symposium I'' , ''Lect. notes in math.'' , '''192''' , Springer  (1971)  pp. 207–253</TD></TR></table>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  V.I. Arnol'd,   "Singularities of caustics and wave fronts" , Kluwer  (1990)</TD></TR></table>