Lecture
8.1. Motivating considerations
One of the central results of singularity theory is the classification of germs of smooth mappings |: В" -+ Е" at critical
(singular) points, i.e. bringing them to the simplest possible form (called
the «normal form») by choosing suitable local
coordinates. It is called a classification because each normal form is a representative of some equivalence class,
the corresponding equivalence relation being that two
germs are carried into one another by a change of local coordinates.
What has been said can be illustrated by special cases which we have already
encountered: in section 4 we brought to normal
form germs of functions (mappings В" - В!), in section 6 germs of plane curves (mappings В! -» ®?). In classifying
germs of functions we used changes of variables only in
the preimage space, whereas in the case of curves we made changes both in the preimage (change of parameter on the curve), and in the image (change of coordinates
on the plane).
In this section we shall give the general notion of equivalence, introducing
the so-called «left-right» equivalence, and also briefly mention some other types of equivalence used in
singularity theory. In the concluding part we shall say a little about
such an important notion as genericity, which often plays the role of a «compass» in choosing the direction of research.
94
8.2. Left-right equivalence
In the general case, when classifying germs of mappings
Е ь Е" _ Е"
two changes of coordinates are used simultaneously: in the preimage space (such changes are called right) and in the image space (they are called left). Right and left changes are chosen
independently of one another. Germs of two mappings are called left-right
equivalent (abbreviated С®-equivalent) if they
are carried into one another by suitable right and left changes.
Let us give a more detailed definition of С®-equivalence for mappings { : В" - ЕВ". Here it will be important for us to distinguish
the preimage space and the image space, even in the case п = т,
so we denote them by the letters М and ЛМ respectively. For the reader
familiar with the notion of a manifold, the following generalization can be given: М and М are manifolds of dimensions ® and т respectively!.
Consider a germ of a mapping
Из): ММ, Л=(Л,..., 1"), (8.1)
and local coordinates 5 = (т1,.... 2.) in М and з = (21,...,2т) in М.
Right and left changes of variables are (local) diffeomorphisms ф : М -} М and ф: М -+’ М respectively, which have
the form
ф: (2) = (7), 9: (=) => (2), (8.2)
where т = (71,..., 2.) and з= (21,...,2т); see fig. 8.1. In the new coordinates 7,2 the mapping / turns into a mapping 9, related to }
by
доф=фор + |9=фоГоф "|. (8.3)
The change ф : № -» М is called right, and the change 2 : М -» М is called
left, in accordance with the order in which ф,? stand in the boxed
equality from (8.3).
DEFINITION 8.1. Germs of mappings
1: ММ, 9:МмМ>мМ (8.4)
1Despite the appearance of a generalized formulation, the
results obtained further will be completely the same for spaces and manifolds, since
all the statements proved below are local.
95
are called С®-equivalent if they are related by relation (8.3) with suitable diffeomorphisms ф : № -+ М and у: М + М.
The СХ-equivalence relation, defined above on the set А of germs of mappings, possesses the properties of reflexivity, symmetry and transitivity. Thus, the quotient set А/^- is defined, whose elements (classes of СЮ-equivalence) are called singularities of mappings of the given type, and the simplest
(in one sense or another) representatives of each class are called
normal forms of germs of mappings.
ы М
Fig. 8.1 Diagram illustrating the definition of left-right
equivalence: changes of coordinates in the image М and preimage ЛГ
REMARK 8.1. In many sources, the term А-equivalence is used to denote left-right
equivalence. But we prefer to use the notation СХ, as more expressive and admitting natural variations. If only the right change ф is used (i.e. setting 12 = {4 — the identity),
one obtains ®-equivalence, and if only the left change 4 is used (i.e.
setting ф = 14), one obtains С-equivalence.
EXAMPLE 8.1. The Morse lemma states that the germ of a smooth
function at a nondegenerate critical point is Ю-equivalent to its
quadratic part. The Tougeron theorem states that the germ of a smooth function at a critical point of arbitrary finite multiplicity д
is -equivalent to its Taylor polynomial of degree д - 1.
The normal forms А„, В) of germs of functions were also obtained by means of only right changes of variables.
96
EXAMPLE 8.2. The normal forms of germs of plane curves (6.6) with
exponents п = 2, 3 also belong to СЮ-equivalence, since
they were obtained using both right and left changes. It is clear that right changes alone
are not sufficient to obtain such normal
forms.
PROBLEM 8.1. Prove that if the germ of a smooth mapping
(8.1) has rank г, then it is ®-equivalent to the germ of a mapping whose г components have
the form of projections:
(т...) = Жь 1Е{И,....ы}.
Recall that the rank of a germ of a mapping is called the rank of its Jacobian matrix at the point under consideration, whereas at points arbitrarily close to it the rank may be equal to or greater than г.
PROBLEM 8.2. Prove that the germ of any smooth mapping (8.1) of constant rank г (here we assume that in some neighborhood of the point under consideration the rank of the Jacobian matrix equals
the same number г) is С®-equivalent to the germ of a mapping with
components
т ЕТ... Г,
0, 1 1=и+1,..., 0. (аа. щь) =
In the special case г = п = т this means that the germ of a diffeomorphism is СЮ-equivalent to the germ of the identity mapping #4.
Prove the stronger statement: the germ of any diffeomorphism is Ю-equivalent to 14, and also С-equivalent to #4.
Let us emphasize that in the definition of С -equivalence it is important that
the right and left changes are chosen independently of one another. If
we used the same changes in the image and the preimage (i.e.
set ф = 4), we would obtain a much more complicated classification. For illustration consider the simplest case — the germ of a
function of one variable {(т) at a noncritical point, which,
as we know, is СЮ®-equivalent (and even Ю-equivalent) to the germ of the simplest function }(т) = т.
PROBLEM 8.3. Show that by means of the change (8.3) with ф=ф
the germ of the function
Г(2) = ах + 62? + о(27), аб 0,
at zero cannot be brought to a linear form. For the proof
it suffices to verify the impossibility of bringing to a linear form
the 2-jet of this function at zero by means of such a change.
97
8.3. Group action
The notion of equivalence of objects of a geometric nature is often based on the choice of some group of transformations of the space whose points constitute these objects. For example, in a course
of analytic geometry one studies various classifications of curves and surfaces of second order: affine, orthogonal, projective. They are based on different notions of equivalence: by
means of affine, orthogonal, or projective transformations (of the plane or space) respectively.
In the case of С®-equivalence of germs of mappings (8.4) there is also
a group of transformations. For simplicity of exposition we shall
consider the germ of a mapping at the origin 0 and assume that {(0) = 9(0) = 0. Denote by the symbols ОЁх and ОЁм the groups
of germs of diffeomorphisms ф : № -+ М and ф: М - М at the origin of the corresponding spaces, such that $(0) = (0) = 0.
Then the notion of Т-equivalence is obviously connected with the group ОШ», the notion of С-equivalence with the group ОЁм, the notion of 27®-
equivalence with their Cartesian product ОШ» х Ом, which is also a group.
Let И/ be an arbitrary set, whose elements we shall denote by Greek letters, and С an abstract group with neutral
element е.
DEFINITION 8.2. We shall say that the group. С’ acts on
И’, if a mapping is defined
А: сх И-М),
satisfying the following two conditions:
1. А(е, <) = & for any & Е ПИ’,
2. А(9. А(п,5)) = А(91,&) for any 9. ВЕСиб ЕЙИ..
Then for each point & Ее И’ one can consider the set
Се '— {А(9,&) : У9 = С},
which is called the orbit of the point & under the action of the group С’. Obviously, any two orbits either do not intersect, or completely
coincide. The union of all orbits covers the entire set И” (by
virtue of the trivial inclusion & Е С).
Let us introduce on the set И” an equivalence relation: & ^> 7, if &, 77 belong to the orbit of some one point, which is equivalent to
98
the condition 1 Е С: or, equivalently, the condition & Е С„. Then the orbits С: are equivalence classes, i.e. elements of the quotient set И// -- (see section 3).
Thus, we obtain a partition of the set И’ into disjoint subsets — equivalence classes, i.e. a classification of the elements of the set И’, generated by the action of the given group А: СхИ’ + И’. For each equivalence class one chooses some representative of it, having, if possible, the simplest «possible» form (the choice of which is sometimes subjective). Such
representatives are called normal or canonical forms of the objects of the set И’ with respect to the chosen equivalence relation. This is the general notion of classification.
For example, СК-equivalence and the corresponding classification of germs of mappings are obtained if one defines the action of the group С = ОШм х Ом on the set И’ by the formula фо роф`! (boxed
in (8.3)). The validity of properties 1 and 2 in definition 8.2 in this
case is completely obvious.
8.4. Contact equivalence
Besides СЮ-equivalence (and its special cases С and ®), there exist other types of equivalence of germs of mappings. In this
introductory course we shall only briefly mention one of them, which is called
contact (К-equivalence).
Let ® be the group of germs of mappings
ф: М - ПИ м,
associating to a point 2 Е М a diffeomorphism Ф(т) е Ом.
We have in this case a mapping
Ф (ту): Мх М+М, Ф(т,0) =0 (УЕ М),
the last equality being a consequence of the definition of the group ОЁм. The group operation in ©’ is defined by
Ф (т, у) у Ф(т, у) — Ф (5, Ф(т, у)),
the neutral element being the germ of the mapping Ф(х,у) = у (УхЕе М).
К-equivalence is obtained by taking the group С = ® х Ом and
defining its action on the set И’ by the formula
А((Ф, 2), 1) = (=, (2) ор", ФЕ, рЕБИ, ГЕИ. (3.5)
99
DEFINITION 8.3. Germs of mappings { and 9 of form (8.4) are called Х-equivalent if
Ф(т, 1(2)) = 9($(т)) (8.6)
for some фе Ом and фе Ом.
Comparing (8.3) and (8.6), it is easy to conclude that С®-equivalence is stronger than ^К-equivalence: the former is obtained from
the latter if in formula (8.5) the group is replaced by its subgroup consisting of such diffeomorphisms Ф(х) е Ом which do not depend
on т. Therefore, if two germs of mappings are СЛХ-equivalent, then they are
К-equivalent, but the converse, in general, is not true (an example will be given below).
The following fact holds: the definition of К-equivalence does not
change if in formula (8.5) the group ®’ is replaced by its subgroup,
consisting of linear diffeomorphisms Ф(т) Е ОШм, i.e. germs of
mappings Ф(т,у) = Л(т)у, where А(х)(.) : В" + В" is a nondegenerate linear operator, smoothly depending on х. We leave the proof
of this to the reader (the exact statement is given in problem 8.4).
PROBLEM 8.4. Prove that if for germs (8.4) relation (8.6) holds with some ф Е ОШх and фе П\Ьу, then there exist
a nondegenerate matrix А(т) of order т, whose elements smoothly
depend on 1 Е М, and a germ ф Е ОШ, such that
А(2) (2) = 9($(2)). (8.7)
Let /+ and [’ be the local ideals of germs [ and д from (8.4). A right
change 2 => Ф(х) carries the ideal Г, into [5, where д = доф. Equality (8.7)
means that [; = Г. It turns out that the following criterion holds:
PROBLEM 8.5. Germs of mappings (8.4) are К-equivalent if and
only if there exists a local diffeomorphism х +} $(х), which carries the local ideal 1 into Г, or conversely.
Let us illustrate the difference between left-right and contact equivalence by an example of germs of mappings В -+ В?. Namely,
the germ of the curve
х=Ри®, у=Е"о(, и(0)ь (0) = 0, (8.8)
is far from always С®-equivalent to the simplest one:
ЕР, у=ЕИ. (8.9)
100
For example, in section 6 we saw that in the special case when
т = п +1, СЮК-equivalence of (8.8) and (8.9) holds only for
values п = 2,3, and for п > 4 it does not.
PROBLEM 8.6. Prove that for any exponents п, пт the germ of
the curve (8.8) is К-equivalent to (8.9).
SOLUTION. This statement follows obviously from problem 8.5, since
the local ideals of mappings (8.8) and (8.9) coincide and are equal
to the principal ideal generated by the monomial 1, where р = шш{п, т}. Moreover, it follows from this that the right change Ён} 2(® in formula (8.7)
can be chosen to be the identity. However, one can also give another proof, not using problem 8.5. We leave this to
the reader to do. и
PROBLEM 8.7. Give an example of two smooth functions ®? - В,
whose germs are К-equivalent, but not С®-equivalent.
REMARK 8.2. In the book (p. 105, item 6.5) К-equivalence
is called У-equivalence, and its definition (equivalent to 8.3)
is given via relation (8.7). Moreover, there exist various
modifications of К-equivalence and С®-equivalence. The reader interested in the details is referred to the articles [17, 27, 31] and
the book [33]. We also note article [13], in which various notions of equivalence and their applications to the study of
singularities are analyzed.
Comments