Foundations of Algebraic Topology

345 Pages·2016·11.073 MB·English

by  Eilenberg, Samuel;Steenrod, Norman

FOUNDATIONS OF ALGEBRAIC TOPOLOGY
PRINCETON MATHEMATICALS ERIES
Editors: MARSTON MOBSE and A. W. TUCKER
1. The Classical Groups, Their Invariantsa nd Representations. By HERMANN
WEYL.
2. Topological Groups. By L. PONTRJAGIN. Translated byE MMA LEHMER.
3. An Introductiont o Differential Geometry with Use of the Tensor Calculus.
By LTJTHER PFAHLEB EISENHART.
4. Dimension Theory. By WITOLD HUBEWICZ andH ENRY WALLMAN.
5. The Analytical Foundations ofC elestial Mechanics. By AUBEL WINTNER.
6. The Laplace Transform.B y DAVID YEBNON WIDDER.
7. Integration. By EDWARD JAMES MCSHANE.
8. Theory of Lie Groups: I. By CLAUDE CHEVALLEY.
9. Mathematical Methodso f Statistics. By HARALD CRAMER.
10. Several Complex Variables. By SALOMON BOCHNERa nd WILLIAM TED MARTIN.
11. Introduction to Topology. By SOLOMON LEFSCHETZ.
12. Topology of Surfaces and Their Transformations. By JAKOB NIELSEN and
WEBNEB FENCHEL.
13. Algebraic Curves. By ROBERT J. WALKER.
14. The Topology of Fibre Bundles. By NORMAN STEENROD.
15. Foundations of Algebraic Topology. By SAMUEL EILENBERG and NORMAN
STEENROD.
16. Functionals ofF inite Riemann Surfaces. By MENAHEM SCHIFFER andD ONALD
C. SPENCER.
17. Introduction toM athematical Logic,V ol. I. By ALONZO CHURCH.
ADNUOF TSNOI
FO
AEGL ARB CI
T YGOLOPO
BY SAMUEL EILEJYBERG
AND
NORMAN STEENROD
PRINCETON, NEW JERSEY
PRINCETON UNIVERSITY PRESS
1952
Copyright, 1952, by Princeton University Press
L. C. Card 52-5841
LONDON: GEOFFREYC UMBERLEGE
OXFORD UNIVERSITY PRESS
PRINTED IN THE UNITED STATES OF AMERICA
THE WILLIAM BYRD PRESS, INC., RICHMOND, VA.
TO SOLOMON LEFSCHETZ
IN ADMIRATION AND GRATITUDE

rP eface
1. PREAMBLE
T eh lapicnirp noitubirtnoc fo siht koob si na citamoixa hcaorppa ot
eht trap fo ciarbegla ygolopot dellac ygolomoh .yroeht tI si eht tsedlo
dna tsom ylevisnetxe depoleved noitrop fo ciarbegla ,ygolopot dna yam
eb dedrager sa eht niam ydob fo eht .tcejbus T eh tneserp noitazitamoixa
si eht tsrif hcihw sah neeb .nevig T eh laud yroeht fo ygolomohoc si
esiwekil .dezitamoixa
tI si demussa taht eht redaer si railimaf htiw eht cisab stpecnoc fo
arbegla dna fo tniop tes .ygolopot oN tpmetta si edam ot ezitamoixa
eseht .stcejbus T sih sah neeb enod ylevisnetxe ni eht .erutaretil O ru
tnemeveihca si tnereffid ni .dnik H ygolomo yroeht si a noitisnart ro(
)noitcnuf morf ygolopot ot .arbegla tI si siht noitisnart hcihw si
.dezitamoixa
S gnikaep ,ylhguor a ygolomoh yroeht sngissa spuorg ot lacigolopot
secaps dna smsihpromomoh ot suounitnoc spam fo eno ecaps otni
.rehtona T o hcae yarra fo secaps dna spam si dengissa na yarra fo
spuorg dna .smsihpromomoh nI siht ,yaw a ygolomoh yroeht si na
algebraic image fo .ygolopot T eh domain fo a ygolomoh yroeht si eht
s'tsigolopot dleif fo .yduts stI range si eht dleif fo yduts fo eht .tsiarbegla
T lacigolopo smelborp era detrevnoc otni ciarbegla .smelborp
nI siht ,tcepser ygolomoh yroeht slellarap citylana .yrtemoeg H­wo
,reve ekilnu citylana ,yrtemoeg ti si ton .elbisrever T eh devired ciarbegla
metsys stneserper ylno na tcepsa fo eht nevig lacigolopot ,metsys dna
si yllausu hcum .relpmis T sih sah eht egatnavda taht eht cirtemoeg
melborp si deppirts fo laitnesseni serutaef dna decalper yb a railimaf epyt
fo melborp hcihw eno nac epoh ot .evlos tI sah eht egatnavdasid taht
emos laitnesse erutaef yam eb .tsol nI etips fo ,siht eht tcejbus sah
devorp sti eulav yb a taerg yteirav fo lufsseccus .snoitacilppa
O ru smoixa era stnemetats fo eht latnemadnuf seitreporp fo siht
tnemngissa fo na ciarbegla metsys ot a lacigolopot .metsys T eh smoixa
era categorical ni eht esnes taht owt hcus stnemngissa evig cihpromosi
ciarbegla .smetsys
2. THE NEED FOR AXIOMATIZATION
T eh noitcurtsnoc fo a ygolomoh yroeht si ylgnideecxe .detacilpmoc
tI si eurt taht eht snoitinifed dna yrassecen sammel nac eb desserpmoc
vii
via PREFACE
within ten pages, and the main properties established within ah undred.
But this is achieved by disregarding numerous problems raised by the
construction, andi gnoring the problem of computing illustrative exam­
ples. These are serious problems, as is well nk own to anyone who has
taught the subject. There is need for a perspective, and a pattern into
which the student can fit the numerous parts.
aP rt of the complexity of the subject is that numerous variants of
the basic definitions have appeared, e.g. the singular homology groups
of eV blen, Alexander, and eL fschet,z the relative homology groups of
eL fschet,z the iVetoris homology groups, the Cech homology groups, the
Alexander cohomology groups, etc. The objectiveo f each variant was to
extend the validity of some basic theorems, and thereby increase their
range of applicability.
In spite of this confusion, a picture has gradually evolved of what is
and should be a homology theory. Heretofore thish as been an imprecise
picture which the expert could use in his thinikng but not in his exposi­
tion. A precise picture is needed. It is at just this stage in the develop­
ment of other fields of mathematics that an axiomatic treatment ap­
peared and cleared the air.
The discussion will be advanced by a rough outline of the construc­
tion of the homology groups of a space. There are four main steps as
follows :
)1( space *— complex
)2( complex >— oriented complex
)3( oriented complex »— groups of chains
)4( groups of chains »— homology groups
In the first form of )1(it was necessary to place on a space the struc­
ture of a complex by decomposing it into subsets called cells, each cell
being a homeomorph of a euclidean cube of some dimension, and any
two cells meeting, if at all, in common faces. It was recogniez d that
only certain spaces, called triangulable, admit such a decomposition.
There arose the problem of characteriizng triangulable spaces by other
properties. This is still unsolved. pS ecial classes of spaces e( .g. differ-
entiable manifolds ) have been proved triangulable and these suffice for
many applications.
The assumption of triangulability was eliminated in three different
ways by the worsk of iVetoris, eL fschet,z and Cech. In each case, the
relation that the complex be a triangulation of the space was replaced
by another more complicated relation, and the complex had to be
infinite. The gain was made at the cost of effective computability.
tS ep )2( has also been a source of trouble. The problem is to assign
THE NEED FOR AXIOMATlZATION ix
integers (incidence numbers) to each pair consisting of a cell and a face
(of one lower dimension) so as to satisfyt he condition that the boundary
of a cell be a cycle. This is always possible, but the general proof
requires the existence of a homology theory. To avoid circularity, it
was necessary to restrict the class of complexes to those for which
orientability could be proved directly. The simplicial complexes form
such a class. This feature ands everal othersc ombined to make simplicial
complexes the dominant type. Their sole defect is that computations
which use thema re excessively long, so muchs o that they are impractical
for the computation of the homology groups of a space as simple as a
torus.
Steps (3) and (4) have not caused trouble. They are purely alge­
braic and unique. The only difficulty a student faces is the absence of
motivation.
The final major problem is the proof of the topological invariance of
the composite assignment of homology groups to a space. Equivalently,
one must show that the homology groups are independent otfh e choices
made in steps (1) and (2). Some thirty years were required for the de­
velopment of a fully satisfactory proof of invariance. Several problems
arising along the way have not yet been solved, e.g. Do homeomorphic
complexes have isomorphic subdivisions?
The origin of the present axiomatic treatment was an effort, on the
part of the authors, to write a textbook on algebraic topology. We were
faced with the problem of presenting two parallel lines of thought. One
was the rigorous and abstract development of the homology groups of a
space in the manner of Lefschetz or Cech, a procedure which lacks ap­
parent motivation, and is noneffective so far as calculation is concerned.
The other wast he nonrigorous, partly intuitive, and computable method
of assigning homology groups which marked the early historical develop­
ment of the subject. In addition the twol ines had to be merged eventu­
ally so as to justify the various computations. These difficulties made
clear the need of an axiomatic approach.
The axioms which we use meet this need in every respect. Their
statement requires only the concepts of point set topology and of
algebra. The concepts of complex, orientation, and chain don ot appear
here. However, the axioms lead one to introduce complexes in order to
calculate the homology groups of various spaces. Furthermore, each
of the steps (2), (3), and (4) is derived from the axioms. These deriva­
tions are an essential part of the proof of the categorical nature of the
axioms.
Summarizing, the construction of homology groups is a long and
diverse story, with a fairly obscure motivation. In contrast, the axioms,