Combinatorial Topology Volume 2

265 Pages·1998·8.277 MB·English

by  P.S. Alexandrov

P.S. Aleksandrov
C O M B I N AT O R I A L
T O P O L O G Y
Volume 2
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COM BIN A TORI A L TOPOLOGY
VOLUME 2
THE BETTI GROUPS
OTHER GRAYLOCIC PUBLICATIONS
KHINCHIN: Three Pearls of Number Theory
PONTRYAGIN: Foundations of Combinatorial Topology
NOVOZHILOV: Foundations of the Nonlinear Theory
of Elasticity
ALEKSANDROV: Combinatorial Topology, Vol. 1
PETROVSKlI: Lectures on the Theory of Integral Equations
KOLMOGOROV and FOMIN: Elements of the Theory of Functions and
Functional Analysis, Vol. 1. Metric and
Normed Spaces
COMBINATORIAL TOPOLOGY
VOLUME 2
THE BETTI GROUPS
BY
P. S. ALEKSANDROV
G R A Y L O C K P R E S S
ROCHESTER, N. Y.
1957
'J’KANSJ^ATK I > J’lfOJVI '1’J I 10 FIRST RUSSIAN KDITION
BY
HORACE KOMM
Copyright, 1957, by
GRAYLOCK PRESS
Rochester, N. Y.
All rights reserved. This book, or parts
thereof, may not be reproduced in any
form, or translated, without permis­
sion in writing from the publishers.
Library of Congress Catalog Card Number 56—13930
Manufactured in the United States of America
CONTENTS
PART THREE
THE BETTI GROUPS
Chapter
VII. Chaixs. Tiie Opehator A
1. Orientation.................................................................................... 2
1.1. Orientation of the space Rn................................................ 2
1.2. Orientation of a simplex and of a skeleton...................... 3
1.3. The body of an oriented simplex...................................... 5
1.4. Extension of an orientation tn to an orientation Rn.
The product orientations tnRn and tinUn.......................... 5
1.5. The orientation (e0C-1)........................................................ 0
1.6. Affine images of orientations.............................................. 7
2. Intersection Number of Planes and Simplexes........................ 8
2.1. Intersection number of planes............................................ 8
2.2. Intersection number of simplexes...................................... 10
2.3. Intersections and simplicial mappings.............................. 11
3. Incidence Numbers....................................................................... 11
3.1. Definition of the incidence numbers................................. 11
3.2. Properties of the incidence numbers................................. 12
4. Cell Complexes; o-complexes...................................................... 13
4.1. Definition of a-complexes and cell complexes.................. 14
4.2. The incidence matrices of a cell complex......................... 16
5. Chains............................................................................................. 18
5.1. Definition of a chain............................................................ 18
5.2. Some remarks on chains...................................................... 20
5.3. Monomial chains. Chains as linear forms........................ 2L
5.4. Chains of a simplicial complex........................................... 22
5.5. The scalar product of chains.............................................. 23
5.6. Extension of chains; restriction of chains to a subcom­
plex. The operators and ........................................... 23
6. The Lower Boundary Operator (The Operator A)................. 24
6.1. Definition of the A-boundary............................................. 24
6.2. Examples of chains and their boundaries......................... 26
6.3. Cycles; chains homologous to zero; the groups ZT($)
and IITm ............................................................................... 29
6.4. Homologies. The symbol Linear independence of
chains with respect to homology....................................... 30
V