Titles in This Series 43 Loi s A. Caffarelli and Xavier Cabre, Full y nonlinear elliptic equations, 199 5 42 Victo r GuiUemin and Shlorao Sternberg, Variation s on a theme by Kepler, 199 0 41 Alfre d Tarski and Steven Givant, A formalization of set theory without variables, 198 7 40 R . H. Brag, Th e geometric topology of 3-manifolds , 198 3 39 N . Jacobson, Structur e and representations of Jordan algebras, 196 8 38 O . Ore, Theor y of graphs, 1962 37 N . Jacobson, Structur e of rings, 195 6 36 W . H. Gottschalk and G. A. Hedlund, Topologica l dynamics, 195 5 35 A . C Schaeffe r and D. C Spencer , Coefficien t region s for Schlicht functions , 195 0 34 J . L Walsh , Th e location of critical points of analytic and harmonic functions, 195 0 33 J . F. Ritt, Differentia l algebra , 195 0 32 R . L Wilder , Topolog y of manifolds, 194 9 31 E . Hille and R. S. Phillips, Functiona l analysis and semigroups, 195 7 30 T . Radd, Lengt h and area, 1948 29 A . Weil, Foundation s of algebraic geometry, 194 6 28 G.T . Whyburn, Analyti c topology, 194 2 27 S . Lefschetz, Algebrai c topology, 194 2 26 N.Levinson , Ga p and density theorems, 194 0 25 Garret t Birkhoff, Lattic e theory, 194 0 24 A . A. Albert, Structur e of algebras, 193 9 23 G . Szego, Orthogona l polynomials, 193 9 22 C . N. Moore, Summabl e series and convergence factors, 193 8 21 J . M. Thomas, Differentia l systems , 193 7 20 J . L . Walsh, Interpolatio n and approximation b y rational function s i n the complex

domain, 193 5 19 R . E. A. C Pale y and N. Wiener, Fourie r transforms in the complex domain, 193 4 18 M - Morse, Th e calculus of variations in the large, 193 4 17 J . M. Wedderburn, Lecture s on matrices, 1934 16 G . A. Bliss, Algebrai c functions, 193 3 15 M L H. Stone, Linea r transformations in Hilbert space and their applications to analysis,

1932 14 J . F. Ritt, Differentia l equations from the algebraic standpoint, 193 2 13 R . L. Moore, Foundation s of point set theory, 193 2 12 S . Lefschetz, Topology , 193 0 11 D . Jackson, Th e theory of approximation, 193 0 10 A . B. Coble, Algebrai c geometry and theta functions, 192 9 9 G . D. Birkhoff, Dynamica l systems, 192 7 8 L . P. Eisenhart, Non-Riemannia n geometry, 192 2 7 E . T.Bell, Algebrai c arithmetic, 192 7 6 G . C Evans , Th e logarithmic potential, discontinuous Dinchlet and Neumann problems,

1927 5.1 G . C . Evans, Functionat e and their applications; selected topics , includin g integral

equations, 191 8 5.2 O . Veblen, Analysi s situs, 1922

4 L . E. Dickson, O n invariants and the theory of numbers W. F. Osgood, Topic s in the theory of functions of several complex variables, 1914

(Continued in the back of this publication)

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Algebraic Topology

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America n Mathematica l Societ y

Colloquiu m Publication s Volum e 27

Algebraic Topology

Solomo n Lefschet z

America n Mathematica l Societ y Providence , Rhod e Islan d

http://dx.doi.org/10.1090/coll/027

1991 Mathematics Subject Classification. Primar y 55-01 , 57-01 .

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14 1 3 1 2 1 1 1 0 9 8 0 2 0 1 0 0 9 9 9 8

CONTENTS CBAtram PAO B

I. INTRODUCTIO N T O GENERA L TOPOLOG Y 1 1. Primitiv e concepts 1 2. Topologica l space s 5 3. Aggregate s o f sets . Coverings . Dimensio n 1 3 4. Connectednes s 1 4 5. Compac t spaces 1 7 6. Separatio n axioms 2 3 7. Invers e mapping systems 3 1 8. Metrizatio n 3 3 9. Homotopy . Deformation . Retractio n 3 9

II. ADDITIV E GROUP S 4 1 1. Genera l propertie s 4 1 2. Generator s of a group 4 9 3. Limit-group s 5 4 4. Grou p multiplicatio n 5 9 5. Characters . Dualit y 6 3 6. Vecto r space s 7 2

III. COMPLEXE S 8 8 1. Complexes . Definition s and example s 8 9 2. Homolog y theory o f finite complexes , (a ) Generalitie s 9 8 3. Homolog y theor y o f finite complexes, (b ) Integra l group s 10 1 4. Homolog y theory of finite complexes, (c ) Arbitrary groups of coefficients. . 10 5 5. Applicatio n to some special complexe s 10 9 6. Dualit y theor y for finite complexe s 11 6 7. linkin g coefficients . Dualit y in the sense of Alexande r 12 4 8. Homolog y theory o f infinite complexe s 12 7 9. Augmentabl e and simple complexe s 13 0

IV. COMPLEXES : PRODUCTS . TRANSFORMATIONS . SUBDIVISION S 13 4 1. Product s of complexe s 13 4 2. Product s of chain s and cycles 13 7 3. Set-transformation s 14 3 4. Chain-mapping s 14 5 5. Chain-homotop y 15 2 6. Complement s 15 9 7. Subdivision . Derivation . Partitio n 16 2

V. COMPLEXES : MULTIPLICATION S AN D INTERSECTIONS . FIXE D ELEMENTS . MANI -FOLDS 17 3

1. Multiplication s 17 3 2. Intersection s 17 8 3. Coincidence s and fixed elements 19 1 4. Combinatoria l manifolds 19 6

VI. NET S O F COMPLEXE S 21 3

1. Definitio n o f net s and their groups 21 3 2. Dualit y an d intersections 21 8 3. Furthe r properties of nets 22 0

CONTENTS

CHAPTXB PAQ B

4. Spectr a 22 9 5. Applicatio n t o infinite complexe s 23 2 6. Web s 23 3 7. Metri c complexe s 23 9

VII. HOMOLOG Y THEOR Y O F TOPOLOGICAL SPACE S 24 4 1. Homolog y theory : foundations an d genera l propertie s 24 4 2. Relation s betwee n connectednes s an d homology 2S 7 3. Group s relate d t o web s 26 2 4. Group s relate d t o th e unio n and intersection o f tw o sets 26 6 5. Th e Vietori s homology theor y fo r eompacta 27 1 6. Reductio n o f th e Vietori s theor y t o th e Cec h theory 27 3 7. Homolog y theorie s of Kurosc h and Alexander-Kolmogorof f 27 8

VIII. TOPOLOG Y O F POLYHEDRA AN D RELATE D QUESTION S 28 7 1. Geometri c complement s 28 7 2. Homolog y theor y 29 3 3. Geometri c manifold s 30 2 4. Continuou s an d singular complexes 30 7 5. Coincidence s and fixed points 31 8 6. Quasi-complexe s an d the fixed point theore m 32 2 7. Topologica l complexe s 32 7 8. Differentiabl e complexe s and manifolds 33 0 9. Grou p manifolds 33 5

10. Nomenclatur e o f complexe s an d manifolds . . 34 1 APPENDIX 34 4

A. O n homology groups of infinite complexes and eompacta. B y Samuel Eilen-berg and Saunders MacLane 34 4

B. Fixe d point s of periodi c transformations . B y P . A . Smit h 35 0 BIBLIOGRAPHY 37 4 INDEX O F SPECIAL SYMBOL S AN D NOTATIO N 38 1 INDEX — 38 3

PREFACE

When the present volume was first contemplated some five years ago it wa s primarily meant to be a second edition of the author's Topology (1930 , Volume XII of the American Mathematical Society Colloquium Series). I t soon became evident howeve r that th e subjec t ha d move d to o rapidl y fo r a mer e revise d edition, and that a completely new book would have to be written. Wit h the consent of the Colloquium Committee the task was undertaken by the author and resulted in the present work. It s basic topic, often referred to as "Combi-natorial Topology," is in substance the theory of complexes and its applications. Many factors have contributed to a great increase in the role of algebra in this subject. Fo r thi s reaso n i t i s mor e appropriatel y describe d a s "Algebrai c Topology," and this explains the title of the volume.

The purel y topologica l (non-algebraic ) par t ha s bee n concentrate d i n th e first chapter , an d al l th e necessar y group-theoreti c materia l i n th e second , thus resulting in a great economy and simplification in the treatment of many questions, notabl y dualit y an d intersections . Th e nex t thre e chapter s dea l with the theory of complexes proper. Th e basic type selected is A. W. Tucker's modified i n that the elements may also take negative dimensions. A s is wel l known one of the important recent advances has been the extension to complexes of th e dualit y an d intersectio n propertie s o f manifolds . Thi s ma y b e ac -complished by means of special "dual" cycles (the "pseudocycles" of Topology, Chapter VI) , o r by a specia l dua l comple x a s don e b y Tucke r (companio n algebraic development by W. Mayer), or else again with Alexander and Whitney without new elements but wit h a new boundary operator for the chains . B y utilizing negative dimensions it has been possible to associate with each complex X a dua l complex X* suc h that the relation between the two is wholl y sym-metrical. A s a consequenc e th e "co-theory " o f X (Whitney' s terminology ) appears a s th e ordinar y theor y o f X* y an d al l th e dualit y an d intersectio n properties are obtained by combining the X, X* relationship with group-duality and group-multiplicatio n i n th e sens e o f Pontrjagin . Ther e .emerges thu s a theory of complexes of purely algebraic nature, with manifolds relegated to the second plane .

The homology theory o f topologica l space s is taken up in Chapter VII, th e necessary limiting processes constituting the theory of nets and webs being dealt with in Chapter VI. W e have chosen as our basic theory the Cech homology theory and in substance reduced to it the other known theories thus unifying a domain which has definitely stood in need of it for some time.

The relativ e concept s which played such an important role in the previou s volume hav e no t bee n neglecte d i n th e present . The y appea r chiefl y i n th e guise of certain binary dissections which run right through complexes, nets and topological spaces, and are at the root of the mechanism of webs.

PREFACE

The last chapte r contains th e applications t o polyhedr a an d certain related questions, notabl y a very concis e and very genera l treatmen t o f fixed points. The book concludes with an appendix by Eilenberg and MacLane on the homol-ogy groups of infinit e complexe s and another by Pau l Smit h o n his theory of fixed points of periodi c transformations .

Owing to limitations o f tim e and space i t has not been possible to tak e up the applications of algebraic topology. Howeve r with Marston Morse's Calculus of Variations in the Large (1934, Volum e XVII I o f th e Colloquiu m Series) , W. V . D . Hodge' s The Theory and Applications of Harmonic Integrals (1941 , Cambridge University Press) , and a forthcoming volum e by Hassle r Whitney on sphere spaces , the reader interested i n th e application s wil l readily satisf y his curiosity.

Certain deviations from standard usage have been adopted in the text an d should be kept i n mind . Thu s "compact " replace s "bicompact, " and "com -plex" replaces "abstract complex." ( A nomenclature of complexe s and mani-folds i s give n a t th e en d o f Chapte r VIII. ) Al l group s ar e topologica l (th e topology may be discrete); unless otherwise stated homomorphisms are supposed to b e continuou s an d group-isomorphism s topological , exception s bein g indi -cated by the mention "in the algebraic sense." Fo r vector spaces over a field there is a special set of conventions indicated in Chapter II (22.2).

The literature i n topology ha s grow n to suc h proportions that i t ha s been impossible to provide more than a scanty bibliography. Reference s are given by th e author' s nam e followe d b y a n appropriat e lette r i n squar e brackets . Those to the present volum e are of th e form (IV , 16.3) , where IV stands for Chapter IV and 16.3 for the numbering in the chapter.

It has been my good fortune to hav e obtained sympathetic cooperation and advice fro m man y sources . I n preparatio n o f th e manuscrip t invaluabl e as-sistance was received from Samuel Eilenberg, W . W. Flexner, N. E. Steenrod, John Tukey, and as regard s the second chapter, Claude Chevalley practically acted as a collaborator. Part s of the manuscrip t in more or less final form or important parts of the proofs were carefully read by Hubert Arnold, E. G. Begle, Paco Lagerstrom , Saunder s MacLane, Mose s Richardson , Seymou r Sherman, J. D. Tamarkin , A. D. Wallace and Hassle r Whitney. T o one and all it i s a great pleasure to express here my appreciation and thanks.

S. LEFSCHET Z PBXNGETON, N . J .

October, 191 1

BIBLIOGRAPHY The following bibliography is rather complete as regards the books published from 1930

onwards. Th e list of papers has been reduced to the strict minimum required in the present work.

LIST O F BOOKS

ALEXANDROFF, P. , an d HOPF , H . [A-HJ Topologie, Berlin, Springer , 1935 , (Die Grundlehren de r mathematischen Wissen -

schaften, bd . 45) . BOUBBAKI, N .

[B] Elements de Mathimatiques. I . Thiorie des Ensembles: I I I . Topologie Ginirale, Paris, Hermann , 1939-1940 , (Actualitl s Scientifiques , nos . 846, 858).

HODGE, W . V . D . [H] The Theory and Application of Harmonic Integrals, Cambridg e Universit y Press ,

1941. HUBEWICZ, W. , an d WALLMAN , H .

[H-W] Dimension Theory, Princeto n Universit y Press , 1941 , (Princeton Mathematica l Series, no . 4).

KUBATOWSKI, K .

[K] Topologie, Warsaw, 1933. LEFSCHETZ, S .

[L] Topology, New York, 1930 , (American Mathematical Society Colloquium Publica-tions, vol. 12).

[Li] Topology, Note s b y N . Steenro d an d H. Wallman , Princeton , 1935 . [I*] Topics in Topology, Princeto n Universit y Press , 1942 , (Annals o f Mathematic s

Studies, no. 10). MENGEB, KAB L

[Me] Kurventheorie, Berlin , Teubner , 1932. MOORE, R . L .

[Mo] Foundations of Point Set Theory, Ne w York , 1932 , (America n Mathematica l Society Colloquiu m Publications , vol . 13).

MOBSE, MABSTON

[Mor] The Calculus of Variations in the Large,'Sew York , 1934, (American Mathematica l Society Colloquiu m Publications , vol . 18).

NEWMAN, M . H . A . [N] Topology of Plane Sets of Points, Cambridg e Universit y Press , 1939.

PONTBJAGIN, L . [P] Topological Groups, Princeto n Universit y Press , 1939 , (Princeton Mathematica l

Series, no . 2). REIDEMEISTEB, K.

[R] Einfuhrung in die combinatorische Topologie, Braunschweig , Vieweg , 1932. [Ri] Topologie der Polyeder, Leipzig , Akademisch e Verlagsgesellschaft , 1938.

SlEBPINSKI, W . [S] Introduction to General Topology, Universit y o f Toronto Press , 1934.

SEIFEBT, H. , an d THBELFALL , W . [S-T] Lehrbuch der Topologie, Leipzig , Teubner , 1934.

TUKET, J . W . [T] Convergence and Uniformity in Topology, Princeton University Press, 1940, (Annals

of Mathematic s Studies , no . 2). 374

BIBLIOGRAPHY 37 5

VEBLEN, O . [V] Analysis Situs, 2 d edition , Ne w York , 1931 , (America n Mathematica l Societ y

Colloquium Publications , vol . 5 , par t 2) . WEFL, A .

[W] UIntegration dans Us Groupes Topologiques et ses Applications, Paris , Hermann , 1941, (Actualit y Scientifiques , no . 869) .

WHYBTJBN, G . T . [Wh] Analytic Topology, Ne w York, 1942, (American Mathematical Society Colloquiu m

Publications, vol . 28) . WILDES, R . L M an d AYBES , W . L. , editors .

[UM] Lectures in Topology, The University o f Michigan Conferenc e o f 1940 , University of Michiga n Press , 1941 .

LIST O F PAPER S

ALEXANDER, J . W . [a] A proof of the invariance of certain constants of analysis situs, Transaction s o f th e

American Mathematical Society , vol . 1 6 (1915) , pp . 148-154 . [b] A proof and extension of the Jordan-Brouwer separation theorem, Transactions o f th e

American Mathematica l Society , vol . 2 3 (1922) , pp. 333-349. [bj Combinatorial analysis situs. I , Transactions o f th e American Mathematical Society ,

vol. 28 (1926) , pp. 301-329. [c] On the chains of a complex and their duals, Proceeding s o f th e Nationa l Academ y o f

Sciences, vol . 2 1 (1935) , pp . 509-511 . [d] On the connectivity ring of an abstract space, Annals of Mathematics, (2) , vol. 37 (1936),

pp. 698-708 . [e] A theory of connectivity in terms of gratings, Annal s of Mathematics, (2) , vol. 39 (1938),

pp. 883-912 . ALEXANDER, J . W. , an d ZIPPIN , LE O

[a] Discrete Abelian groups and their character groups, Annal s o f Mathematics , (2) , vol . 36 (1935) , pp. 71-85.

ALEXANDROFF, P . S . [a] Untersuchungen tiber Oestalt und Lage abgeschlossener Mengen beliebiger Dimension,

Annals of Mathematics , (2) , vol . 30 (1928) , pp. 101-187. [b] Dimensions theorie. Ein Beitrag zur Geometric der abgeschlossenen Mengen, Mathema-

tische Annalen , vol . 10 6 (1932) , pp . 161-238 . [c] On local properties of closed sets, Annal s of Mathematics, (2) , vol. 36 (1935), pp. 1-35 . [d] Diskrete Rdume, Matematicheski i Sborni k (Recuei l Mathematique) , (n . s.) , vol . 2

(1937), pp . 501-318 . [e] Zur Homologie-Theorie der Kompzkten, Compositi o Mathematica , vol . 4 (1937) ,

pp. 256-270 . [f] General combinatorial topology, Transactions o f th e American Mathematical Society ,

vol. 4 9 (1941) , pp . 41-105 . ALEXANDROFF, P . S. , an d PONTRJAGIN , L .

[a] Les vari6Us d n dimensions geniralisees, Compte s Rendus de l'Acadeinie des Sciences, Paris, vol. 202 (1936) , pp. 1327-1329.

ALEXANDROFF, P . S. , an d URYSOHN , P . [a] Memoirs sur lesespaces topologiques compactes, Verhandelingen, section I , Akademi e

van Wetenschappen, Amsterdam , vol . 1 4 (1929) , pp . 1-96 . BEGLE, E .

[a] Locally connected spaces and generalized manifolds, American Journal of Mathematics, vol. 6 4 (1942) .

BORSUK, KABO L [a] Sur les rUractes, Fundament a Mathematicae , vol . 1 7 (1931) , pp. 152-170 .

376 BIBLIOGRAPHY

BBOTJWEB, L . E . J . [a] On continuous vector distributions on surfaces, Proceedings , Akademi e va n Weten -

schappen, Amsterdam, vol . I I (1909) , pp. 850-858; vol. 1 2 (1910) , pp. 716-734; vol. 13 (1910) , pp . 171-186 .

[b] On continuous one-to-one transformations of surfaces into themselves, Proceedings , Akademie va n Wetenschappen , Amsterdam , vol . 1 1 (1909) , pp . 788-798 ; vol . 1 2 (1910), pp . 286-297; vol. 1 3 (1911) , pp. 767-777; vol. 1 4 (1911) , pp. 300-310; vol. 1 5 (1913), pp. 352-360; vol. 2 2 (1920) , pp. 811-814; vol. 2 3 (1921) , pp . 232-234.

[c] Beweis der Invariam der Dimensionemahl, Mathematisch e Annalen , vol . 7 0 (1911) , pp. 161-165.

[d] Vber Abbildungen von Mannigfaltigkeiten, Mathematisch e Annalen , vol . 7 1 (1912) , pp. 97-115 .

[e] Beweis der Invariant dea n-dimenaionalen Gebiets, Mathematisch e Annalen , vol . 7 1 (1912), pp . 305-313 .

[f] Vber die periodischen Transformationen der Kugel, Mathematisch e Annalen , vol . 8 0 (1919), pp . 39-41 .

BBOWN, A . B. , an d KOOPMAN , B. C . [aJ On the covering of analytic loci by complexes, Transactions of the American Mathemat-

ical Society , vol . 3 4 (1932) , pp . 231-251 . CxiBNa, S . S .

[a] On the triangvlation of regular loci, Annal s o f Mathematics , (2) , vol . 3 5 (1934) , pp . 579-587.

[b] Triangulation of the manifold of class one, Bulleti n o f th e America n Mathematica l Society, vol . 4 1 (1935) , pp . 549-552.

CECH, EDUARD [a] Theorie generate de Vhomologie dans un espace quelconque, Fundamenta Mathematicae,

vol. 1 9 (1932) , pp. 149-183. [b] Theorie generate des varietes et de leurs theoremes de dualite, Annal s o f Mathematics ,

(2), vol . 3 4 (1933) , pp . 621-730 . [c] Sur lea nombrea de Betti locaux, Annals of Mathematics, (2) , vol. 35 (1934), pp. 678-701. [d] Lea groupea de Betti oVun complexe infini, Fundament a Mathematicae. vol . 25 (1935) ,

pp. 33-44 . [e] Lea theoremes de dualite en topologie, Casopi s pr o Pestovan f Matematik y a Fysiky ,

vol. 6 4 (1935) , pp . 17-25 . [f] Multiplications on a complex, Annals of Mathematics, (2) , vol. 37 (1936) , pp. 681-697. [g] On bicompact spaces, Annal s o f Mathematics , (2) , vol. 38 (1937) , pp. 823-844.

CHEVALLET, CLAUD E [a] Sur la definition dea groupea de Betti des ensembles fermes, Compte s Rendus de 1'Aca-

d&nie des Sciences, Paris, vol. 20 0 (1935) , pp. 1005-1007. DOWTEB, C . H .

[a] Hopfa theorem for non-compact spaces, Proceeding s o f th e Nationa l Academ y o f Sciences, vol . 2 3 (1937) , pp . 293-294 .

ElLENBEBG, SAMUE L [a] Cohomolagy and continuous mappings, Annal s o f Mathematics , (2) , vol . 4 1 (1940) ,

pp\ 231-251. [b] On a theorem of P. A. Smith concerning fixed points for periodic transformations, Duk e

Mathematical Journal , vol . 6 (1940) , pp . 428-437. EILENBEBG, SAMUEL , an d MACLAN E SAUNDEBS

[a] Group extensions and homology, Annals o f Mathematics , (2) , vol. 4 4 (1943). FLEXNEB, W . W .

[a] On topological manifolds, Annal s o f Mathematics , (2) , vol . 3 2 (1931) , pp . 393-406 ; The Poincark duality theorem for topological manifolds, ibid., pp. 539-548.

BIBLIOGRAPHY 377

[bj Character group of a relative homology group, Annal s o f Mathematics , (2) , vol . 4 1 (1940), pp . 207-214 .

[c] Simplicial intersection chains for an abstract complex, Bulletin of the American Mathe-matical Society , vol . 4 6 (1940) , pp . 523-524.

FREUDENTHAL, HAN S

[a] Alexanderscher und Gordonscher Ring und ihre Isomorphic, Annal s o f Mathematics , (2), vol . 3 8 (1937) , pp . 647-655 .

[b] Entwicklungen von Raumen und ihren Gruppen, Compoeiti o Mathematica , vol . 4 (1937), pp . 145-234 .

[c] Bettische Gruppe mod. 1 und Hopfsche Gruppe, Compositio Mathematica, vol. 4 (1937) , pp. 235-238 .

[d] Die Triangulation der differemierbaren Mannigfaltigkeiten, Proceedings , Akademi e van Wetenschappen, Amsterdam, vol. 42 (1939) , pp. 880-901.

GELFAND, L. , an d RAIKOV , D . [a] On the theory of characters of commutative topological groups, Compte s Rendu s d e

l'Acad&nie de s Science s d e l'URS S (Doklady) , (n.s.) , vol. 2 8 (1940) , pp . 195-198 . GORDON, I .

[a] On intersection invariants of a complex and its complementary spaces, Annal s of Mathe-matics, (2) , vol . 3 7 (1936) , pp. 519-525.

HOPP, H .

[a] Zur Algebra der Abbildungen von Mannigfaltigkeiten, Journa l fu r die rein e und ange-wandte Mathematik , vol . 16 3 (1930) , pp . 71-88.

[b] Beitrdge zur Klassifizierung der Fldchenabbildungen, Journa l fu r di e rein e un d ange-wandte Mathematik , vol . 16 5 (1931) , pp . 225-236 .

[c] Vber die Topologie der Gruppenmannigfaltigkeiten und ihre Verallgemeinerungen, Annals o f Mathematic s (2) , vol. 4 2 (1941) , pp. 22-52.

HUREWICZ, W . [a] Beitrdge zur Topologie der Deformationen. I . Hoherdimensionale Homotopiegruppen,

Proceedings, Akademi e va n Wetenschappen , Amsterdam , vol . 3 8 (1935) , pp . 112 -119; II. Homotopie- und Homologiegruppen, vol. 38 (1935), pp. 521-528; III. Klassen und Homologietypen von Abbildungen, vol . 3 9 (1936) , pp. 117-126 ; IV. Aspharische Rdume, vol . 3 9 (1936) , pp . 215-224 .

JOHNSON, L . W . [a] A linear algebraic theory of complexes, Princeton thesis , 1941.

KEREKJART6, B . [a] Vber die periodischen Transformations der Kreisscheibe und Kugelfldche, Mathe -

matische Annalen , vol . 8 0 (1919) , pp . 36-38 . KOLMOGOROFF, ANDR E

[a] Les groupes de Betti des espaces locilement bicompactes, Comptes Rendus de l'Acadlmie des Sciences , Paris , vol . 20 2 (1936) , pp . 1144-1147 ; ProprieUs des groupes de Betti des espaces localement bicompactes, ibid. , pp . 1325-1327 ; Les groupes de Betti des espaces metriques, ibid. , pp . 1558-1560 ; Cycles relatifs. Thborhne de dualiU de M. Alexander, ibid., pp. 1641-1643 .

[b] Vber die Dualitdt im Aufbau der kombinatorischen Topotopie , Matematicheskii Sborni k (Recueil Math&natique) , (n.s.) , vol . 1 (1936) , pp . 97-102 .

[c] Homologiering des Komplexes und des lokal-bicompaklen R&umes, Matematicheski i Sbornik (Recuei l Math&natique) , (n.s.) , vol . 1 (1936), pp. 701-705.

KOOPMAN, B . C , an d BROWN , A . B . See Brow n an d Koopma n [a] .

KttNNETH, D . [a] Vber die Bettischen Zahlen einer Produktmannigfaltigkeit, Mathematisch e Annalen ,

vol. 9 0 (1923) , pp . 65-85 ; Vber die Torsionszahlen von ProduktmannigfaUigkeiten, Mathematische Annalen , vol. 91 (1924) , pp. 125-134.

378 BIBLIOGRAPHY

KUBOSCH, ALEXANDE R [a] Kombinatorischer Aufbau der bikompakten topologischen Rdume, Compositi o Mathe -

matica, vol . 2 (1935) , pp . 471-476 . LEFSCHETX, 8 .

[a] Intersections and transformations of complexes and manifolds, Transaction s o f th e American Mathematical Society , vol . 2 8 (1926) , pp. 1-49 .

[b] On singular chains and cycles, Bulletin of the American Mathematical Society , vol . 39 (1933), pp. 124-129 .

[c] On generalized manifolds, America n Journa l o f Mathematics , vol . 5 5 (1933) , pp . 469-504.

[d] On locally connected and related sets. I . Annal s o f Mathematics , (2) , vol . 3 5 (1934), pp. 118-129.

[ej Chain-deformations in topology, Duke Mathematica l Journal , vol . 1 (1935) , pp. 1-18 . [f] On chains of topological spaces, Annals of Mathematics, (2) , vol. 39 (1938), pp. 383-396. (g) On the fixed point formula, Annal s o f Mathematics , (2) , vol . 3 8 (1937) , pp . 819-822 . [h] The role of algebra in topology, Bulletin o f th e American Mathematical Society , vol .

43 (1937) , pp . 345-359 . LEFSCHETZ, 8. , an d WHITEHEAD , J. H . C .

[a] On analytical complexes, Transaction s o f th e America n Mathematica l Society , vol . 35 (1933) , pp . 510-517 .

LEVIN, MADELIN [a] An extension of the Lefschetz intersection theory, Revist a d e Ciencias , vol. 3 9 (1937) ,

pp. 93-118 . MACLANE, SAUNDERS , an d EILENBEBG , SAMUE L

See Eilenber g an d MacLan e [a] . MATER, W .

[a] Vber abstrakte Topdogie, Monatsheft e fu r Mathemati k un d Physik , vol . 3 6 (1929) , pp. 1-42 ; 219-258.

[b] TopologiMche Qruppensysteme, Monatsheft e fu r Mathemati k un d Physik , vol . 4 7 (1938), pp. 40-86,

[c] Charktersysteme und DualiUUstheoreme, Journal o f Mathematics an d Physics , Massa -chusetts Institut e o f Technology, vol . 1 8 (1939), pp. 1-27 .

NEWMAN, M . H . A . [a] On the foundations of combinatory analysis situs, Proceedings , Akademie va n Weten -

schappen, Amsterdam, vol. 29 (1926), pp. 611-626; vol. 29 (1926), pp. 627-641; vol. 30 (1927), pp . 670-673 .

[b] Intersection complexes. I . Combinatory theory, Proceedings o f th e Cambridg e Philo -sophical Society , vol . 2 7 (1931) , pp . 491-501.

[c] A theorem on periodic transformations of spaces, Quarterl y Journa l o f Mathematics , Oxford Series, vol. 2 (1931) , pp. 1-8 .

POINCAR*, H . [a] Analysis situs, Journa l de l'ficole Polytechnique , Paris , (2) , vol. 1 (1895), pp. 1-123 . [b] Complement a Vanalysis situs, Rendiconti , Circol o Matematico , Palermo , vol . 1 3

(1899), pp. 285-343. PONTBJAGiN, L .

fa] Vber den algebraischen Inhalt topologiecher DualiuUssdtzc, Mathematische Annaten , vol. 10 5 (1931) , pp . 165-205 .

[b] The theory of topological commutative groups, Annal s o f Mathematics , (2) , vol . 3 5 (1934), pp . 361-388 .

[c] The general topological theorem of duality for closed sets, Annal s o f Mathematics , (2), vol . 3 5 (1934) , pp . 904-914.

[d] Products in complexes, Matemattcheski i Sborni k (Recuei l Math&natique) , (n.s.) , vol. 9 (1941) , pp . 321-330 .

BIBLIOGRAPHY 379

PONTEJAGIN, L. , an d ALEXANDBOFF , P . S . See Alexandra! ? an d Pontrjagi n [a] .

RAIKOV, D. , an d GELFAND , L . See Gelfan d an d Raiko v [a] .

DE RHAM , GEORGE S

[a] Sur Vanalysis situs des variites d n dimensions, Journa l d e Mathlmatique s Pure s e t Appliques, (9) , vol . 1 0 (1931) , pp . 115-200 .

[b] Relations entre la topologie et la theorie des inUgrales multiples, Enseignemen t Mathe*-matique, vol . 3 5 (1936) , pp . 213-228 .

[c] Sur la theorie des intersections et les inUgrales multiples, Commentari i Mathematic i Helvetici, vol . 4 (1932) , pp . 151-157 .

RICHABDSON, M. , an d SMITH , P . A . [a] Periodic transformations of complexes, Annal s o f Mathematics , (2) , vol . 3 9 (1938) ,

pp. 611-633 . SAMELSON, HAN S

[a] Beitr&ge sur Topologie der Gruppen-Mannigfaltigkeiten, Annal s o f Mathematics , (2) , vol. 42 (1941) , pp. 1091-1137.

SMITH, P . A . [a] A theorem on fixed points for periodic transformations, Annal s o f Mathematics , (2) ,

vol. 35 (1934) , pp. 572-578. [b] Transforniations offiniU period, AjmsisotMa&hemati 127-164 . [c] The topology of transformation groups, Bulletin of the American Mathematical Society ,

vol. 44 (1938) , pp. 497-514. [d] Transformations of finite period II , Annal s o f Mathematics , (2) , vol . 4 0 (1939) ,

pp. 690-711. [e] Transformations of finite period III , Newman's theorem, Annal s o f Mathematics ,

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[b] Regular cycles of compact metric spaces, Annal s o f Mathematics , (2) , vol . 41 (1940), pp. 833-851 .

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(2), vol . 3 6 (1935) , pp . 448-463 .

380 BIBLIOGRAPHY

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[a] On duality and intersection chains in combinatorial analysis situs, Annals of Mathe-matics, (2) , vol. 88 (1932), pp. 521-524.

[b] On C*-complexes, Annals of Mathematics, (2) , vol. 41 (1940), pp. 809-824. WHITEHEAD, J . H . C , an d LEFSCHBTS , 8 .

See Lefschet s an d Whitehead [a] . WHITNEY, HASSLE*

[a] Differentiable manifolds, Annal s of Mathematics , (2) , vol. 37 (1936), pp. 645-680. [b] The imbedding of manifolds in families of analytic manifolds, Annals of Mathematics,

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vol. 3 (1937) , pp . 3545 . [d] On products in a complex, Annals of Mathematics, (2), vol. 39 (1938), pp. 397-432. [e] Tensor products of Abelian groups, Duk e Mathematical Journal , vol . 4 (1938) , pp.

495-528. WILDER, R . L .

[a] Generalized closed manifolds in n-space, Annal s of Mathematics , (2) , vol . 35 (1934), pp. 876-903.

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ADDED IN PROOF

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[d] A new homology theory, Annals of Mathematics, (2) , vol. 43 (1942), pp. 370-380. (ej A new homology theory. II , Annals of Mathematics, (2) , vol. 43 (1942).

INDEX OF SPECIAL SYMBOLS AND NOTATIONS

0,1 C, 3 , 1 { 1, 1

u,u.,n,n.,i

a, 13 aA» , a v « , 1 3 a x © , 13

<S,,119 $,152 if, 9 9 & , U 9 £",99 $,,119 3,68 3«,88 ?,42 9t,88 St, 324 <B(x,«), 34 3 ' , 99 3 , , H 9

a(x), 89 0(p),88 V , 7 , , H 9 r ' , r „ , i i 9 *•», 178 *x", 214 n, 93 i n 1,94

e,», 1 {|},2 / (negation) , 2 - , 2 < , > , 4 {«;>}. 4 +-•,85

o* 9 2 M, W , 246 r, r* 147 «b(J7),*,(F)>248 {«x; T£}, 248

1,9 |C| ,240 | |C ||, 240 C\C, ,119 <*(*,»), 33 £,93 Ga,88 G/H, 45 (?[mj, G(m), G*(m), 68 X,92 1^1,98 M", 197 N(E, P), 63 P", P", 11,12 ie", ioo ff„, 116 t, i*, r\ 14 5 tt, 10 2 £ , 11 6

r|z,2 To,Ti,Tt,Ti,2i

®,87 [:], 89 ~ , 9 9 s^, **\ t l oa = ,19 3 o,*,207

'x,165 X', X 0 0 ,164 X*,91 1,197 X. ,130 Xi X • • • X X ., 135 s-y, 178 X — F 1 {Xx;xi'},214

CI, 89 CS1, CS2, CS3, 7 diam, 34 dim, 14 Dl, D2 , D3, D4, 33 F,99 Int A, 8 KI, 110 Lk, 124 OS1, OS2, OS3, 5, 6 PI, P2,1 7 St, 89

C.229 F, 229 F,229 H,229 Z, 229

P ,3 P",47

381

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INDEX

(This index does not cover the Appendices)

Acyclic, 111, 248 Addition of subsets in groups, 41 Adherence, 295 Aggregates

finite, 1 3 infinite, 1 3 locally finite, 13 neighborhood-finite, 1 3 point-finite, 13

Alexander, 41 . 88 , 124 . 127 , 173 , 174 , 181 . 189. 204 . 227 , 228 . 244 . 255 . 282 , 285 . 304. 309. 310, 315

Alexandroff, 1 , 28, 84, 85 , 88 , 125 , 134 , 139 , m 173 . 213, 216. 244. 255. 266, 287. 305 Analytical

complex, 333 manifold, 331 spread, 333 system of functions , 330

Annihilator, 60 Annul, 60 Antecedent

of a n Euclidean complex, 97 of a singular complex, 313

Augmentation, 130, 221, 248 Axioms

first countability , 6 Hausdorff, 24 second countability, 6 To, T u T i9 T if 2 4 triangle, 33 for metric spaces, 33 for topologica l spaces , 5

Barycentric mapping and induced chain-mapping, 291 subdivision, 169

Base, 6 canonical, 104 closed, 7 countable, 6 equivalent, 6 of a group, 50 of a vector space, 73 at a point , 6

Basic class, 29 set, 20

Begte. 322, 343 Betti

group, 102 number, 100,107,116, 216, 249

Bicompact, 17 Borsuk, 40, 292 Boundary

of a chain, 98 of a simplex, 112 of a subset of a topological space, 9 in a complex, 89,90

Bounded, totally, 34 Bourbaki. 19 Brouwer, 196 , 299, 304, 305, 321, 342, 343 Brown, 333 Cairns, 287. 330. 333. 335 Carrier

of a chain-mapping, 152 of intersection s i n a complex , 17 9 in a topological space , 255

Cartan, 285, 335, 340 O-concepts, 330 C"-concepts, 331 Cech, 1, 79, 128. 173, 189, 213, 216, 244, 252.

254, 273,285,342,34 3 Cell, 11

continuous, 306 oriented, 305 parametric, 331 singular, 311 singular oriented, 312

Chain, 15,48,98,113,229,271,282,31 2 Chain-deformation, 152,160,242,31 3 Chain-derivation, 165,31 3

reciprocal of , 16 6 Chain-graph, 149 Chain-homomorphism, 48, 242 Chain-homotopy, 152,160 , 242, 313

prismatic, 155 Chain-mapping, 145 , 159, 242, 313 Chain-retraction, 154 , 242

384 INDEX

Chain-subdivision, 163 reciprocal of , 16 3

Chain-transformation, 145,15 9 Character, 63, 80

group, 63 space, 80

Characteristic number , 94 Chevalley, 41 , 73, 213, 244 Circuit, 114

absolute, 114 modular or relative, 115 non-orientable, 115 orientable, 11 5 simple, 115 simplicial, 115 invariance under derivation, 167 topological invariance , 301

Class. see homology clas s intersection, 181,207,22 0 Kronecker index , 120 , 181 , 207, 220

Closure in a complex, 89 in a space, 9

Coboundary, 116 Cochain, 116, 229, 283 Cocyle, 116 , 216, 229, 242, 271, 283 Cofinal, 5 Cohomology, 116

ring, 181 Coincidences, 191,210,318,32 4 Coinitial, 5 Combinatorial cycles , classes, etc., 293 Compact spaces, 17,18

countably, 17 linearly, 78 locally, 22 locally linearly, 79 sequentially, 38 product of , 1 0

Compacting, 20 Compactum, 17 , 37 Complement in a set, 1 Complete metric space, 35 Complexes, 89

augmentable, 130 augmented, 130 C , analyti c differentiate , 33 3 closure-finite, 91 complete Vietoris , 271 continuous, 308 derived, 164 , 167 dual, 91, 93 Euclidean, 96 infinite, 127 , 232 locally finite, 91

metric, 239 polyhedral, 9 3 reciprocal, 19 7 simple, 13 2 simplicial, 9 2 singular, 312 star-finite, 9 1 topological, 327 Vietoris, 271 nomenclature of , 34 1

Component, see connectednes s Conet, 214 Connectedness, 14 , 90, 110 , 217, 257 Continuous transformation , 7 Convention for vector spaces , 72 Convergence, 2 5

of cycles , 263 Coset, 45 Covering, 13

closed, 13 irreducible, 247 open, 13 shrinkable, 26 strongly irreducible, 247

Cycles, 99, 215, 229, 240, 271, 282 absolute, 113 relative or modular, 113 around a set, 263 through a set, 262

Cyclic, 111 , 248

Deformation, 3 9 cell, 308 theorem of Alexander-Vebien , 309

Degree o f a mapping , 321 Dense, 9

c dense, 37 De Rham , 204, 285 Derivation, 16 4 Derived complexes , 164 , 167

barycentric, 17 0 Diameter, 34, 239 Difference

of sets , 1 in groups , 41

Differentiable, 33 0 Dimension, 14, 74,89, 299,300 Direct syste m o f groups , 57, 75 Directed syste m o f sets , 4 Discrete topology , 8 Dissection

of a complex, 112 of a net, 223 of a space, 248

INDEX 385

Distance, 33 Division-closure group , 68 Dowker, 282 Dual

categories, 121 complex, 91,9 3 pairing, 66,81 systems, 67 weakly dual systems, 83

Duality theorem s absolute, 117,121 , 129 , 219, 230, 233, 248,

303 relative, 122 , 203 , 224 , 248 of Alexande r type, 124,129 , 204, 227, 255,

304 of Poincarl , 203 for combinatorial manifolds , 202 for geometric manifolds, 303 , 304 for groups, 66, 81 for nets , 218, 224, 227, 230, 233 for topological spaces , 248,250, 255 for webs , 236

c aggregates, etc., 34, 37, 39 Eilenberg, 128 , 311 Essential cycle, 216 Essential element , 7 9 Euler characteristic , 10 4 Euler-Poincarl formula , 104 Extension

theorem of Tietze , 28 of continuous functions, 28 of fields, 85 of transformations , 2

/- , jE-chain-mapping, 15 9 Face, 90 Factor-group, 45 Field, 72

character, 80 extension, 85

Finite intersection property , 17 Fixed elements of a chain-mapping , 191 Fixed point , 318 , 324

property, 326 Flexner. 174. 188. 342 Free group, 50 Freudenthal, 41,181, 213 Function, 2

locally finite, 28 3 Fundamental group , 310

zero-cocycle, 131 , 221

r-complex, 336

r-manifold, 33 6 r-space, 341 Generators of a group, 49 Geometrical elements , 293 Gordon, 174,18 1 Graph. 2. 4

of a chain-mapping, 149 of multiplication in a complex, 175

Grating, 279 spectrum, 279

Group, 41, 47 manifold, 335 multiplication, 5 9 space, 336 see als o homolog y group s

Hausdorff, 258 Hausdorff space . 6. 24 Hausdorff-Lennes separatio n condition , 1 5 Heegaard, 335 Hilbert parallelotope , 1 2 HLC* space, 322 £T-net» J7-conet. 235 Hodge. 286. 287 Homeomorphism, 7 Homology, 98

class, 99 group, 99 group o f produc t o f compac t Hausdorf f

spaces, 265 group of product of complexes, 139 group of produc t of nets , 222 ring, 207

Homomorphism, 44 in the algebraic sense, 44 linear, 72

Homotopy, 39 operator, 152

Hopf. 28. 88. 134. 139. 266. 287. 335. 337. 340 Hurewicz. 300. 311. 336

t-chain-mappiiig, 159 Ideal elements , 234 Identification, 3

topological, 12 Imbedding, 3

topological, 1 3 Incidence matrices, 101 Incidence numbers,89,92,95,97,312,327,32 9 Index, see Kronecker index Indexed system, 3 Indicatrix, 305 Induced chain-mappings , 162,18 5 Injection, 48 , 57, 113 , 225, 234, 251

386 INDEX

Interior, 8 point, 0

Intersections in combinatoria l manifolds , 20 5 in complexes , 178 in differentiable manifolds , 834 in infinit e complexes , 18 8 in nets , 218 , 224 in a produc t o f combinatoria l manifolds ,

208 in a produc t o f complexes , 187 in sets, 1 in simplicia l complexes , 188,18 9 in topologica l spaces , 255 of singula r element s in a geometric mani -

fold, 316 induced, 18 4 and chain-mappings , 184 and subdivision , 18 6 carriers of , 179 , 255 components of , 17 8 topological invarianc e of , 294,30 5

Interval, 8 Invariant factors , 5 1 Inverse syste m

of groups , 54 of spaces , 31 of vecto r spaces , 75

Inverse mappin g sequenc e o f spaces , 33 Isomorphism i n complexes , 9 1

metric, 242 weak, 91 in groups , see homomorphis m

Join, 92,137 , 138,14 3 Jordan-Brouwer theorem , 304

Kernel of a homomorphism , 4 5 of a simplex and o f its dual , 245,246

Kline, 244 Kolmogoroff, 125,173,189,244,282 , 285 Koopman, 333 Kronecker deltas , 8 8 Kronecker index , 110,118,178,202,208,21 8

topological invarianc e of , 294,30 6 Kronecker product , 87 Kunneth, 134,13 9 Kuratowski, 9 Kurosch, 244 . 278. 279

Lagerstrom, 25 4 LC space , 322 Lebesgue number, 14 , 37

Lefechetz, 72 . 88, 121,134.139,152,154, 173, 178, 188 , 196 , 203 , 204 , 213 , 244 , 278 , 287, 293 , 308 , 309 . 315 , 316 . 318 , 322 , 333, 335, 342

Linear graph , 9 3 Limit-groups, 54 l imits , 25 limit-space, 31 Iindeldf's theorem , 6 Linear

topology, 7 4 variety, 77

Linking coefficient , 124 , 202,228

MacLane, 128 Manifolds

absolute, 197 combinatorial, 19 6 C-differentiable, analytic , 331 doubly covering , 205 elementary, 201 geometric, 293 , 302 imbedding and coverin g theorems, 333 non-orientable, 198 , 204 orientable, 198 relative o r modular , 19 7 topological, 331 weak, 204 product of , 20 1 with boundary , 197 with regular boundary, 198 coincidences in , 21 0 duality in, 202,303 intersections in , 205 Kronecker index in, 202,208 and chain-mappings , 210 nomenclature, 342

Mapping, 7 barycentric, 291 system, inverse , 31

Matrix, 50 Mayer, 88 , 266 Menger, 14, . 299, 300 Mesh, 34,240,287 Metric

chain-deformation, 24 2 chain-deformation retract , 242 chain-homotopy, 24 2 complex, 239 isomorphism, 24 2 product, 3 5 set-transformation, 24 2 of a space , 33

Metrizable, 34

INDEX 387

Metrization, 34 Moore, 5 Multiplication, 5 9

group, 59 linear, 72 natural, 66 ring, 173 in a complex , 173

Neighborhood, 9 -finite covering , 13 groups o f a close d set , 262

Nerve, 244 Nets, 213

augmented, 22 2 partial, 220 sequential, 214 simple, 214 simplicial, 214 components of , 21 7 products of , 22 2 dissection in , 223

Newman, 173 , 293, 342 Non-orientable

circuit, 11 5 manifold, 198 , 204

Normal couple, 120 space, 26 subcomplex o f a n Euclidea n complex , 292

Normality, 26 Nucleus, 42

Order, orderin g relatio n 4 proper, 4 reflexive, 4

Ordered syste m o r set , 4 partially, 4 simply, 4

Ordering b y inclusion , 4 Orientable

circuit, 11 5 manifold, 198 , 204

Orientation function, 8 9 in a complex , 89, 93 of a simplex , 92

Oriented cell, 30 5 manifold, 30 5

Orthogonal groups , 60

p-section o f a complex , 90

Pairing dual, 66, 81 of groups , 59

Parallelotope, 1 1 compact, 19 Hilbert, 1 2

Partial net, 220 system o f a n inverse system, 82

Partition, 16 7 elementary, 16 8 -equivalence, 293

Permanence relations, 61,62,63, 185,220 Phragmln-Brouwer theorem , 30 5 Poincarl, 88. 99. 203. 293. 327. 335. 342

duality theorem , 20 3 group, 310 polynomial, 104,141 , 222

Polyhedron, 94 continuous, 308

Pontrjagin. 41.42.43.59.63.66 . 68.121.173, 174, 204. 277. 335. 341

duality theore m fo r groups , 66, 81 duality theore m fo r spheres , 304 ring, 341

Prism, 307 Prismatically related , 15 4

in th e strong sense, 154 Product

metric, 35 symmetric, 4 topological, 10 weak produc t o f groups , 47 of chains and cycles, 137 of compac t spaces , 19 of complexes , 134 of groups , 47 of manifolds , 201 of sets, 3 relation to set-transformations an d chain-

mappings, 161 homology groups , Bett i numbers , tonio n

coefficients, 139,222,26 5 intersections in , 187 , 206 projection in , 4

Projection in complexes , 113 in groups , 45, 54, 57 in limit-spaces, 31,32 in nets , 225 in products , 4 in topologica l spaces , 251 in webs, 234

388 INDEX

Projective elements and groups in a spectrum, 229 plane, 3 space, 3

Pseudo-component, 25 8 Pseudo-manifold, 11 5

Quasi-complex, 322 Quasi-component, 258 Quasi-dual categories , 203

Range o f a transformation , 2 Rank o f a group, 50 Realization, Euclidean, 97 Reciprocal complex , 19 7 Reciprocation, 19 8 Refinement

of a covering , 1 3 directed by , 14

Region, 8, 34 Regular

Euclidean complex , 28 9 open or closed sets, 278 point of polyhedron , 302

Reidemeister, 287 Relativization principle , 1 0 Reorientation conventio n fo r chains , 10 0 Retract, 40

neighborhood, 40 Retraction, 40

deformation, 40 Riesz,9

Saraelson, 265, 341 Savage, 68v

Segment, 11 Seifert, 88 , 287 Separable, 35 Separating class , 29 Separation axioms , 2 3 Set-subdivision, 16 3 Set-transformation, 14 3

dual, 145 generalized simplicial , 144 inverse, 145 metric, 242 simplicial, 14 4

Similar aggregates , 13 Similitude, 1 3 Simple

circuit, 115 complex, open or closed, 132

Simplex, 92 Euclidean, 96

boundary of , 9 3 groups of , 11 2

Singular chain-deformation, 31 3 chain-derivation, 313 chain-homotopy, 313 elements (complexes , cycles, groups) , 311 groups related t o a point , 313 intersections, 316 locus of a polyhedron , 302 point o f a polyhedron , 302

Solenoid, 31 Space

compact, 1 7 complete, 3 5 connected, 14 discrete, 8 Euclidean, 8 Hausdorfif, 24 HLC, 222 LC*,m metric, 33, 34 metrizable, 33 normal, 26 ordered, 8 projective, 3 separable, 35 Tt, T u T« , T i9 2 4 topological, 5 Tychonoff o r completely regular , 29

Span,73 Spectrum, 214,22 9 Sphere, 11 , 34, 321

as boundary of a simplex, 93, 112,196,19 9 Spheroid, 34 Spread, analytical, 333 Star

in a complex , 89 in an aggregate of sets , 334

Star condition, 309 Star-finite complex , 91 Steenrod. 1 . 32, 41, 68, 88. 105.106.128. 213,

244 Subbase, 6

at a point , 6 nuclear, 42

Subcomplex, 90 closed, 90 continuous, 309 open, 90

Subdivision, 162 bary centric, 169 invariance under , 163 , 164 , 166 , 167 , 186,

195,200

INDEX 389

Subdivision-—Continued chain-, 163 set-, 16 3

Subnets, 223 closed, 224 complementary, 224 open, 224 injection, projection of, 22 5

Summation conventions, 88, 198

Threlfall, 88 , 287 Tietze's extension theorem. 28 Topological

complex, 327 equivalence, 8 group, 41 space, 5 sphere, 11

Topological invarianc e of circuit, 301 dimension o f a polyhedro n o r Euclidea n

space, 300 geometric manifold, 302 homology theor y (Bett i numbers , torsio n

coefficients, homolog y an d cohomolog y groups, intersections, Kronecker index) in finite an d infinit e polyhedra , 294 , 295, 297, 298

intersections i n geometric manifold , 30 5 Kronecker index in geometri c manifolds ,

306 regionally i n Euclidea n space , 30 0

Topology, 8 discrete, 8 linear, 74

Toroid, 20 Torsion

coefficient, 10 2 group, 102

Transfinite induction , 5 , 50 Transformation

one-one, into , onto, single-valued, multi -valued, n-valued, univalent , 2

inverse, 3 open, closed , continuous , topological* 7 unimodular, 52

Tucker, 88 . 89, 130 , 131 , 134 , 143 , 150 , 154 . 157, 162 , 173. 195. 196. 286

Tukey, 1 , 38, 324 Tychonoff, 19 , 28 Tychonoff o r completel y regula r space , 2 9

Unimodular transformation , 5 2 Union, 1 Univalent transformation , 2 Universal

coefficient group , 109,128 theorem for fields, 106 , 128 , 219, 236, 249

Urysohn, 1 , 14, 299 characteristic function , 2 7 lemma, 27 metrization theorems, 38

van Kampen, 41, 66, 69 Values, of a transformation, 2 Varietv, linear , 77 Veblen, 88, 108 . 173 , 202, 287, 309. 311. 315.

327. 330, 342 F-elements, 240 Vertex, 90 Vietoris. 31. 240, 244. 266. 271. 273 Vietoris elements and theory, 271

Wallace, 1 , 261, 263 Wallman, 1 , 20, 252. 300 Weak duality, 83 Webs, 233, 262 Weil, 41, 66 Whitehead, 173 , 287, 330, 333 Whitney, 69 . 87, 88, 116 . 132 , 133 . 173 , 178.

179, 189 , 190 , 196 , 287, 330, 333, 335 Wilder, 293, 305. 342

Zippin, 41 Zorn'8 lemma, 5