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Smooth Manifolds and Mappings
A Preliminary Review of Some Calculus
The Differential of a Smooth Mapping
Vector Fields and Flows
Germs of Smooth Mappings
The Notion of Transversality
The Basic Transversality Lemma
An Elementary Transversality Theorem
Thom's Transversality Theorem
First Order Singularity Sets
The Finite Dimensional Model
Groups Acting on Sets
Some Geometry of Jets
Smooth Actions of Lie Groups on Smooth
Singular Points of Smooth Functions
Some Basic Geometric Ideas
Determinacy of Germs
Classification of Germs of Codimension
Stable Singularities of Smooth Mappings
The Basic Ideas
Classification of Stable Germs
Higher Order Singularity Sets
Classifying Germs under,11-Equivalence
Some Examples of Classifying Stable Germs
Singular Points of Stable Mappings
The Theorem of Sara
Semialgebraic Group Actions
The Borel Lemma
Guide to Further Reading
Suppose you take a smooth curved surface
made of some transparent material
and imagine it projected downwards onto a plane surface of light from above. on the surface
Think of this as a map
X projected down to a point
Y by shining a beam
f : X - Y
q = f(p)
with every point in the plane
the plane you will see the apparent outline of the surface, as it would appear from below.
Here are two simple examples.
It is not hard to see which points on X
give rise to the apparent outline
they are precisely the points where the tangent plane to the surface is vertical, the so-called singularities of the mapping
f : X - Y.
Figure 1 repre-
sents the simplest situation one can imagine, with the surface X folding over at
are called fold points.
Figure 2 represents a
more complex situation, a curve of fold points on which lies an exceptional point
where two folds meet, a so-called cusp point.
cated situation is provided by Figure 3 where
A still more compli-
is a torus,
nut - shaped surface.
Here one has again curves of fold points, on which lie
Fig. 3 four cusp points:
but in addition we have two simple crossing points on the
apparent outline, where the curves cross over properly. any point
very close to
Thus, if we take
on the apparent outline and look at the nature of the outline we can distinguish just three possibilities.
Common sense, and a certain amount of experimentation, will soon convince one that these are the only essential types of behaviour which can arise, in the sense that any other type of behaviour could be eliminated by the slight-
est change of position of X in space.
For instance our torus might be so
positioned that the apparent outline was as in Figure 4, with the outline touching itself at some point:
but clearly we could just nudge
to get back to the previous situation where only possibilities I, II, III can occur.
To make the point even clearer, here are three further types of behaviour
which are all inessential, because they could be eliminated by the slightest change in position of simple crossing points
to yield situations exhibiting only fold, cusp and
The broad objective of this book is to introduce the reader to the less technical aspects of a mathematical theory of singularities which seeks to make precise the kind of heuristic reasoning just described.
X, Y are replaced by smooth manifolds, which are natural generaliza-
tions to higher dimensions of the familiar notions of curve and surface.
the projection of the surface onto the plane is replaced by an arbitrary smooth mapping
f : X-+ Y.
For such mappings we introduce the general notion of
singularity and begin to list the simplest singularities which can occur,
We shall, for reasons of simplicity, con-
idealizing each type by a model.
cern ourselves principally with the local behaviour of the mapping, behaviour very close to a single point in the domain:
thus in the situation
discussed above we interest ourselves solely in the fold and cusp points, and neglect simple crossing points which arise from considering what happens close to two distinct points in the domain.
There is nothing particularly new in the notion of a singularity.
tists and geometers have recognized them, and appreciated their significance, for a long time now.
But no-one seems to have systematically set about
studying the singularities of smooth mappings till the pioneering work of Hassler Whitney in the mid 1950's.
Around the same time Rend Thom pointed
out the analogy with more finite-dimensional situations and indicated the general lines along which a theory might proceed.
So it was in the 1960's
that a number of mathematicians, principally John Mather, laid the foundations of a general theory.
That was the position in 1967 when Vladimir Arnol'd put
together the bits and wrote his now classic survey paper, a model of lucid descriptive writing.
It was a time of great promise.
itself threw up a number of provocative problems, and the range of possible applications (both within and without mathematics) added to the excitement. Without question, the intervening years have justified that promise, and singularity theory can hold its own as a flourishing area of mathematics.
I feel that the time has come to provide prospective students with readable introductions to the subject.
It is my personal conviction that the way to
get into any area of mathematics is to concentrate on understanding the simplest situations first, so as to build up some intuitive feeling for what is going on, and to leave the deeper matters till later in life.
of Smooth Mappings is the result of following this guiding philosophy.
taken a small number of intuitively appealing ideas and used them to pursue the problem of listing singularity types, one of the goals of the local theory.
It is the kind of book which I would expect a postgraduate student in mathematics to read with little difficulty, and I rather hope that others will find it within their compass as well.
A guide to further reading has been inclu-
ded to help the reader pursue those matters which interest him most. A few words are in order concerning the structure of the book.
dance with the philosophy outlined above smooth manifolds are introduced in
Chapter I as subsets of in enjoying certain properties: way everyone should meet them.
I think this is the
Anyone who wants to get to grips with singu-
larity theory should be familiar with the basic ideas of transversality and of unfolding, so these topics provide the subject matter for Chapters II and III respectively.
Here again I have kept to the simplest situations which can
arise, imposing restrictions whenever I felt it was possible to suppress undue technicality:
in particular, unfoldings have been introduced in a finite-
dimensional situation where they are much easier to understand.
theory proper is taken up in Chapter IV with the study of functions;
enables one to make some distance fairly easily, without getting involved in the subtleties associated to general mappings.
The result is the derivation
of the list of singularities of codimension : 5. case of mappings is taken up:
In Chapter V the general
it is inevitable that one must quote more and
prove less, but I have tried to expose the less technical aspects and give a fairly coherent account of just how one uses the theory to obtain explicit lists of the simplest singularities which turn up.
Of course one can pursue 5
the listing process much further than is indicated in this book, but one soon comes up against much deeper matters which lie beyond the scope of an introductory account.
I have not attempted systematically to attribute results to their authors, mainly on the ground that such a practice is out of place in a book at this level.
In any case, the material of the first four chapters is now pretty
well an established part of the subject.
I should say however that the open-
ing sections on differential topology follow closely the exposition given by John Milnor in his excellent little book "Topology, from the Differentiable The material in the final chapter, basically Mather's classi-
fication of stable germs by their local algebras, is not as well-known as it should be.
Here I decided to follow the elegant account of Jean Martinet
(see the Guide to Further Reading) in which the unfolding idea plays the central role.
The key result in this development, namely the characteriza-
tion of versal unfoldings, turns on a real version of the Weierstrass Preparation Theorem which I do not discuss;
I felt it was more important at this
level to place proper emphasis on the underlying geometric ideas, and to leave an exposition of the Preparation Theorem to a volume with more ambitious aims. I decided also to say nothing about the applications of singularity theory, mainly because I feel each area of application is probably worthy of a volume in itself.
For instance, Thom's catastrophe theory is already the subject
matter of several volumes.
Also, the applications within mathematics itself
all seem to be at too early a stage to merit writing-up. that my guide to further reading will prove to be useful.
Liverpool February 1978
Here again I hope
To Les Lander who drew the pictures,
and helped me find enthusiasm at a time when it was all but lost.
To Peter Giblin who undertook the considerable task of correcting the manuscript,
and whose suggestions have contributed much to the final form of this book.
And to Ann Garfield who did the typing and produced an excellent job in difficult circumstances.
Smooth manifolds and mappings
A Preliminary Review of Some Calculus
[In this section it will be understood that
]RP, 7Rq Let
U, V, W
are open sets in
respectively.] f : U --> V be a mapping with components
f1, ..., f
smooth when all the partial derivatives
(a = a 1 + ... + an )
exist and are continuous in fice to observe that if
For the purposes of this book it will suf-
f1, ..., f
are all given by polynomials in
x1, ..., xn U
will be smooth.
And we call
V when it is a bijection, and both
f : U -+ V
7Rn - ]RP
a diffeomorphism of are smooth.
For every point
there is a
a E U
called the differential of
is precisely the linear mapping whose matrix (relative to the standard bases) is the so-called Jacobian