Table Of ContentLecture Notes ni
Mathematics
Edited by .A Dold and .B Eckmann
795
yrtemoeG dna ygolopoT
III Latin American School of Mathematics
Proceedings of the School held the at
Instituto de Matematica Pura Aplicada e
CNPn ,,~;,~,,oI h.h, 1976
pin ,,4,
detidE yb
Jacob Palis and Manfredo do Carmo
Springer-Verlag
Berlin. Heidelberq • New York 19 7 7
Editors
Jacob Palls
Manfredo do Carmo
Instituto de MatemAtica
Pura e Aplicada
Rua Luiz de Camoes, 68
2000 Rio de Janeiro - R ,J Brasil
AMS Subject Classifications (1970): 49 XX, F 53 XX, 55 XX, 57 XX, 58 XX,
70XX
ISBN 3-540-08345-6 Springer-Verlag Berlin • Heidelberg • New York
ISBN 0-387-08345-6 Springer-Verlag New York • Heidelberg • Berlin
This work is subject to copyright. All rights are reserved, whether the whole
or part of the material is concerned, specifically those of translation, re-
printing, re-use of illustrations, broadcasting, reproduction by photocopying
machine or similar and means, storage ni data banks.
Under § 54 of the German Copyright Law where copies are made for other
than private a use, fee is to the payable the publisher, amount of the fee to eb
with the publisher. agreement by determined
© by Springer-Verlag Berlin • Heidelberg 7791
Printed ni Germany
Printing dna binding: Offsetdruck, Hemsbach/Bergstr. Beltz
012345-0413/]412
Preface
The III Latin American School of Mathematics (III ELAM) aimed
to stimulate the development of the areas of geometry and iopolo~ in
Latin America and to expand the interchange of ideas and contacts among
mathematicians of this region initiated with the support of the 0AS in
the I and II Schools, held respectively in Brazil and Mexico. The IlI
ELAM congregated more than 250 mathematicians (researches and students)
most of them from Latin America. The members of the Organizing Committee
were: C. Camacho, P. Schweitzer, M. do Carmo and J. Palis (Coordinator).
These Proceedings reflect one of the two main activities of the
meeting, namely a series of research talks covering topics in Dynamical
Systems, Differential Geometry, Foliations, Singularities of Mappings
and Algebraic Topology. The other activity was a series of introductory
courses in these subjects given by M. Peixoto, M. do Carmo, P.Schweitzer,
J. Sotomayor and J. Adem, whose lecture notes are being or have been in-
formally published by the Organizing Committee.
We hope that these Proceedings can contribute to the further
progress of these areas of Mathematics within and outside Latin America.
To many of our colleagues, to several universities and research
institute of Latin America and to the director of the host institution
L. Dias we express our gratitude for their most valuable help.
We acknowledge the financial support of the Brazilian agencies
CNPq, FINEP, FAPESP and CAPES, as well as the 0AS.
Jacob Palis and Manfredo do Carmo
Rio de Janeiro, April, 1977
CONTENTS
Jos6 Adem and Evaluation of some Maunder cohomology
Kee Yuen Lam operations
Jorge Solo Andrade Constructions g6ometrique de certaines 32
series discretes
J.L. Arraut and The point spectrum of the adjoint of an 56
N.Moreira dos Santos automorphism of a vector bundle
Thomas Banchoff and Whitney duality and singularities of pro- 68
Clint McCrory jections
Cesar p Camacho and Orbit preserving diffeomorphisms and the 82
A. Lins Neto stability of Lie group actions and singu-
lar foliations
A.G. Colares and On minimal immersions with parallel nor- 104
Manfredo .P do Carmo mal curvature tensor
Shiing-shen Chern Circle bundles 114
D.B,A. Epstein A topology for the space of foliations 132
D.B.A. Epstein and Stability of compact foliations 151
H. Rosenberg
.S Feder and Secondary operations in K- theory and 161
W. Iberkleid the generalized vector field problem
R. G6rard On holomorphic solutions of certain kind 176
of Pfaffian systems with singularities
Herman Gluck and Scattering problems in differential 191
David Singer geometry
.C Gutierrez and The connected components of Morse-Smale 230
W. de Melo vector fields on two manifolds
Gilbert Hector Feuilletages en cylindres 252
Michael .R Herman )I Mesure de Lebesgue et hombre de 172
rotation
)2 The Godbillon-Vey ~nvariant of foli- 294
ations by planes of T
.M Herrera On the construction of the trace in 308
Serre duality
Peter Hilton Localization theories for groups and 319
homotopy types
Henrich Kollmer On hyperblic attractors of codimension 330
one
A. Lins, W. de Melo On Li6nard's equation 335
and C.C, Pugh
.S Lojasiewicz On the Weierstrass preparation theorem 358
IV
Ricardo Ma~ )I Reduction of semilinear parabolic 361
equations to finite dimensional C4flows
)2 Axiom A for endomorphisms 379
)3 Characterizations of AS diffeomor- 389
phisms
Airton .S de Medeiros Structural stability of integrable 395
differential forms
W. de Melo Accessibility of an optimum 429
.J Moser )I The scattering problem for some par- 441
ticle systems on the line
)2 Proof of a generalized form of a fixed 464
point theorem
Jacob Palis Jr. Some developments on stability and bi- 495
furcations of dynamical systems
Lucio .L Rodr~guez Convexity and tightness of manifolds 510
with boundary
Otto Raul Ruiz M. Existence of brake orbits in Finsler 542
mechanical systems
Jose Carlos de On the complex projective bundle con- 568
Souza Kiihl struction
Floris Takens Symmetries, conservation laws and 581
variational principles
Geovan Tavares dos Classification of generic quadratic 605
Santos vector fields with no limit cycles
Juan Tirao The classifying ring of SL(2,C) 641
Friedrich Tomi On the finite solvability of Plateau's 679
problem
A.J. Tromba The set of curves of uniqueness for 696
Plateau's problem has a dense interior
C.T.C. Wall Geometric properties of generic differen- 707
tiable manifolds
E.C. Zeeman A catastrophe model for the stability 775
of ships
Eduard Zehnder )I Note on smoothing symplectic and 828
volume preserving diffeomorphism
)2 A simple proof of a generalization 855
of a theorem by C.L. Siegel
EVALUATION OF SOME MAUNDER COHOMOLOGY OPERATIONS
By
Jos6 Adem and Kee Yuen Lam
Introduction
In this paper we present an explicit evaluation of primary, secon-
dary and tertiary Maunder cohomology operations on complex projective
spaces. These operations are related to divisibility properties of the
Chern character, as was first discovered by Adams for primary operations
([2]) and later extended by Maunder to higher order operations ([14]).
If ~ is the canonical complex line-bundle over the N-dimensional
complex projective space CP N, we have
ch(n_l)n = (em_l)n = n + Gt=l{n+t'n}~n+t'
where the rational numbers {n+t,n} are the Taylor coefficients and
E H2(cpN;z) is the integral cohomology generator. According with
Adams (loc. cit.) the number
(*) m(t){n+t,n}
is an integer, where the numerical function m(t) is defined by
)t(er = ~p[t/(p-1)]
P
where p runs over the primer numbers.
Roughly speaking, to compute the higher order Maunder operations
(mod )2 on the class n is equivalent to determine the highest power
of two contained in the integers (~). In section 3 we give explicit
expressions that allow us to compute the 2-integers 2t{n+t,n} mod 8
and this is enough for our purpose.
We obtain simple formulas in terms of binomial coefficients to
n
evaluate the Maunder secondary and tertiary operations on the class
(see 4( )6 (4.17)) Their duals denoted by ~(2) for secondary and by
2r
• ' " '
)3(r2~ for tertiary operations, are also evaluated (see (6.5), (7.12)).
The secondary operations ~(2) have been used to solve several problems.
2r
Through "accompany relations" they can be combined to yield non-trivial
operations on real projective spaces, that are applied to solve non-
immersion problems ([5],[62). We do not know how to apply this proce-
dure to the ~2r )3( " In general, examples of non-trivial tertiary oper-
ations on real projective spaces promise to be interesting.
We do not give any application of our results, with exception of
this one, presented right here. Let q = 2 n+2 and r n, = 2 with n >_ .2
Our evaluation of tertiary operations in (7.12), gives
)3( q = ~q+r door( 2),
~2r
in H (cPq+r/cPq-I;Z2). Now, a result of Gitler and Milgram ([11]),
(3)
shows that the operation ~2r on an integral class of dimension ~ ~ 2r-7
is a primary operation. This operation in our case becomes zero and we
have proved that the stunted projective space cPq+r/cp q-1 cannot be
desuspended 2q-2r+7 times.
Finally, we want to indicate that Dennis Phee Hurley has obtained
in [16], evaluation results for a full family of modified Maunder dual
operations of all orders. His work agrees with our theorem (7.]2) and
can be regarded as a generalization of the results given in [10], for
secondary operations. However, it seems that he cannot obtain our
theorem (4.]7).
.I The Maunder operations
As defined in [14;p.753], let C(3,r) be the chain complex 3( ! r)
3 d 2 d I d
(1.1)
3 C 2 > C I > C ~"' CO,
where each i is a C free-graded left module over the mod 2 Steenrod
algebra A, on generators described as follows:
O C with A-basis {Co},
I C with A-basis {ci, ci0, c11},
2 C with A-basis {c2, c20, c21, c22},
3 C with A-basis {c3},
where dim O = c ,0 dim i = c 2r+i-1 for i = 1,2,3, and dim Cni = 3n-2i,
for n = 1,2. On these generators, the maps i d are given by
I dlC = (Xsq2r)co ,
(1 .z) dlClO = Sq01Co,
dlCll = sqlco .
d2c 2 = sqlc t + (Xsq2r)c]l + (Xsq2r-2)clO ,
= Sq 1O
d2c20 clO,
(1.3)
+ Sq 10
d2c21 = sqlc10 c11
d2c22 = Sqic11 ,
(1.4) d3c 3 = sqlc2 + (×sq2r)c22 + (Xsq2r-2)c21 + (Xsq2r-4)c20
where Sq 01 = Sq2Sq I + SqISq 2 and × : A > A is the canonical
antiautomorphism. Each map i d is of degree zero and d2d 3 = dld 2 = 0
follows from the relations
(1.5) sqlsq 1 = O,
(1.6) sqO1sq01 = O,
(1.7) sqlsq 01 + sqO1sq 1 = O,
(1.8) SqI(xsq 2r) + (Xsq2r-2)Sq 01 + (Xsq2r)sq I = 0.
Associated with C(3,r) we have a pyramid of operations ~s,t, for
3 > s > t > 0, where each ~s,t is of (s-t) th order. We recall that
m
this set of operations verify the Maunder's axioms and that they sat-
isfy the relations with the Chern character as stated in [14;Th.2].
Let w E H2(cpN'z2 ), be the mod 2 cohomology generator of the
N-dimensional complex projective space, where N is a large enough
number, and let n w be the n-fold product of w. Sometimes it is
necessary to work in a (2n-1)-connected complex (see [14;Th.I~). In
this case, through the collapsing map, n w is regarded as an element
of H (cpN/cpn-I;z2). Set
(1.9) s : O C + H , (cpN/cp n-l' .z2) '
as the A-map defined by s(c0) = n w For s (or equivalently for w n)
we shall make explicit the primary, secondary and tertiary operations
associated with the chain complex (1.1).
In order to motivate the computation developed in section ,3 we
advance the following. Let q be the canonical complex line bundle
over CP N, and ~ = q-1. The n-fold product n can be regarded as an
element of the Grothendieck ring Kc(CpN/cpn--I ) (see [3;p.622]). If
E H2(cpN;z) represents the integral cohomology generator then, for
the Chern character, we have chH = em-1 and
(1.10) chp n = (e~-1) n.
.2 A binomial identity
In this section we state the Jensen identity concerning binomial
coefficients (P). Let a,B,y and n be positive integers. Then we have
(2.1) n (a Bk k), ry+B n-k (n_k) ) = ~ n gi(~+Y+n~'-l)"
k=0 i=0 n -1
For a proof see [9].
.3 Computation of 2t{n+t,n} mod 8
Write (e~-1) n : E {n+t,n)a n+t where {n+t,n} are the Taylor
t=0
coefficients. The value of 2t{n+t,n} mod r 2 must be computed in order
to evaluate the Maunder operations on CP ~. In this section we compute
them mod .8 Computation mod 16 is also available. In general the task
becomes more complicated modulo higher powers of .2
All computations and divisibility arguments take place in Z(2 ,)
the ring of rational numbers with odd denominators. If a,b E Z(2),
a K b (mod 2 )r means that a-b = 2rc for some c E Z(2 .) This is the
only kind of congruence considered in our paper.
