Selected Works, Volume 3: Algebraic and Differential Topology

275 Pages·2021·27.254 MB·English

by  L. S. Pontryagin

L. S. PONTRYAGIN
SELECTED WORKS
Volume 3
Algebraic and Differential Topology
Classics of Soviet Mathematics
L. S. PONTRYAGIN SELECTED WORKS
Edited by R. V. Gamkrelidze
Volume 1: Selected Research Papers
Volume 2: Topological Groups
Volume 3: Algebraic and Differential Topology
Volume 4: The Mathematical Theory of Optimal Processes
ISSN 0743-9199
L. S. PONTRY AGIN
SELECTED WORKS
Volume 3
Algebraic and Differential Topology
Translated from the Russian by P. S. V. Naidu
Boca Raton London New York
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First published 1986 by Gordon and Breach Science Publishers
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ISBN 13: 978-2-88124-134-5 (set)
ISBN 13: 978-2-88124-035-5 (hbk) (volume 3)
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Part One of this book was originally published in Russian as OCH0Bbl K0M01-1H8T0pH0lil
Tononornlil and Part Two as rnaAK1-1e MHorooOpa31,tff Iii li!x nplilMeHeHlilff e Teopli!i-1
roMOToni-1111 by Izdatel'stvo Nauka, Glavnaya redaktsiya fizikomatematicheskoi literatury,
Moscow, both in 1976.
Library of Congress Cataloging-in-Publication Data
Pontryagin, L. S. (Lev Semenovich), 1908-
Algebraic and differential topology.
(L. S. Pontryagin selected works ; v. 3) (Classics
of Soviet mathematics, ISSN 0743-9199)
Bibliography: p.
I. Algebraic topology. 2. Differential topology.
I. Gamkrelidze, R. V. II. Series: Pontryagin, L. S.
(Lev Semenovich), 1908- . Selections. Polygot.
1985 ; v. 3. III. Series: Classics of Soviet
mathematics.
QA3.P76 1985 vol. 3 [QA612) 5IO s 85-9793
[514'.2]
ISBN 2-88124--035--6 (Switzerland : v. 3)
Lev Semenovich Pontryagin

Contents
Editor's Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xt
FOUNDATIONS OF ALGEBRAIC TOPOLOGY
Preface to the Second Russian Edition . . . . . . . . . . . . . . . . . . . . . 3
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
Notation............................................... 7
CHAPTER I: COMPLEXES AND THEIR HOMOLOGY
GROUPS................................ 9
1. Euclidean Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2. Simplex. Complex. Polyhedron . . . . . . . . . . . . . . . . . . . . . . . . 18
3. Application to Dimension Theory. . . . . . . . . . . . . . . . . . . . . . 25
4. Homology Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
5. Subdivision into Components. Zero-dimensional Homology
Group.............................................. 38
6. Betti Numbers. Euler-Poincare Formula . . . . . . . . . . . . . . . . 41
CHAPTER II: INVA RIANCE OF HOMOLOGY GROUPS 49
7. Simplicial Mappings and Approximations . . . . . . . . . . . . . . . 50
8. Conic Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
9. Barycentric Subdivision of a Complex . . . . . . . . . . . . . . . . . . 62
10. Lemma on the Covering of a Simplex and its Applications 68
11. Invariance of Homology Groups Under Barycentric Sub-
division. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
12. Invariance of Homology Groups. . . . . . . . . . . . . . . . . . . . . . . 77
CHAPTER III: CONTINUOUS MAPPINGS AND FIXED
POINTS................................. 85
13. Homotopic Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
14. Cylindric Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
15. Homological Invariants of Continuous Mappings . . . . . . . . 97
16. Theorem on the Existence of Fixed Points............... 103
Vll
Vlll CONTENTS
SMOOTH MANIFOLDS AND THEIR
APPLICATIONS IN HOMOTOPY THEORY
Preface to the Second Russian Edition . . . . . . . . . . . . . . . . . . . . . 117
Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
CHAPTER I: SMOOTH MANIFOLDS AND THEIR
SMOOTH MAPPINGS................... 121
1. Smooth Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
Concept of a smooth manifold. . . . . . . . . . . . . . . . . . . . . . . . . 121
Smooth mappings.................................... 124
Certain methods of constructing smooth manifolds. . . . . . . 126
2. Embedding of a Smooth Manifold into a Euclidean Space. 130
Smooth mapping of a manifold into a manifold of greater
dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
Projection operation in a Euclidean space . . . . . . . . . . . . . . . 131
Embedding theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
3. Improper Points of Smooth Mappings.................. 137
Putting into general position . . . . . . . . . . . . . . . . . . . . . . . . . . 138
Sard's theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
4. Nondegenerate Singular Points of Smooth Mappings . . . . . 143
Typical points of self-intersection under a mapping from a
manifold Mk into a vector space E2k................ 145
Typical critical points of a numerical function on a manifold 148
Typical irregularities of a mapping from a manifold Mk into
a vector space E2k-I . . . . . . . . . . . . • . . . . . . . . . . . . . . . . . 152
Canonical form of typical critical points and typical irregular
points.......................................... 156
CHAPTER II: FRAMED MANIFOLDS................. 159
5. Smooth Approximations of Continuous Mappings and
Deformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
Structure of the neighbourhood of a smooth submanifold 159
Smooth approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
6. The Main Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
Framed manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
Passage from mappings to framed manifolds............. 167
Passage from framed manifolds to mappings.. . . . . . . . . . . . 171
CONTENTS IX
7. Homology Group of Framed Manifolds . . . . . . . . . . . . . . . . 175
Homotopies of framed manifolds . . . . . . . . . . . . . . . . . . . . . . 175
Homology group II~ of framed manifolds . . . . . . . . . . . . . . . 178
Orthogonalization of frames. . . . . . . . . . . . . . . . . . . . . . . . . . . 180
8. The Suspension Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
CHAPTER III: THE HOPF INVARIANT................. 187
9. Homotopic Classification of Mappings from n-Dimensional
Manifolds into ann-Dimensional Sphere.. . . . . . . . . . . 187
Degree of a mapping................................. 187
Mappings from an n-dimensional sphere into an n-dimen
sional sphere. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
Mappings from an n-dimensional manifold into an n-dimen
sional sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
10. The Hopflnvariant of a Mapping from the Sphere L2k+l into
the Sphere sk+ 1• • • • • . • • • • • • • • • • • • • • • • • • • • • • • • • . • . 194
Looping coefficient. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
The Hopf invariant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
The Hopf invariant of a framed manifold . . . . . . . . . . . . . . . 197
11. Framed Manifolds with Vanishing Hopf Invariant. . . . . . . . 201
Reconstruction of manifolds. . . . . . . . . . . . . . . . . . . . . . . . . . . 202
Manifolds with vanishing Hopf invariant. . . . . . . . . . . . . . . . 206
CHAPTER IV: CLASSIFICATION OF THE MAPPINGS
OF (n + 1)- AND (n + 2)-DIMENSIONAL
SPHERES INTO AN n-DIMENSIONAL
SPHERE................................ 211
12. Group of Rotations of a Euclidean Sphere . . . . . . . . . . . . . . 211
Quarternions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
Covering homotopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
Group of rotations of a Euclidean space . . . . . . . . . . . . . . . . 215
13. Classification of the Mappings from a Three-dimensional
Sphere into a Two-dimensional Sphere.............. 219
Mappings of a sphere into a circle . . . . . . . . . . . . . . . . . . . . . 220
The Hopf mapping from a three-dimensional sphere into a
two-dimensional sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
Classification of the mappings from a three-dimensional
sphere into a two-dimensional sphere . . . . . . . . . . . . . . . 223