Topics in Topology. (AM-10), Volume 10 9781400882335

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ANNALS OF MATHEMATICS STUDIES NUMBER 10

TOPICS IN TOPOLOGY BY

SOLOMON LEFSCHETZ

PRINCETON PRINCETON UNIVERSITY PRESS L O N D O N : H U M P H R E Y M ILF O R D O XFO RD U N IV E R S IT Y PRESS

1942

Copyright 1942 P r in c e t o n U

n iv e r s it y

P ress

Lithoprinted in U.S.A.

EDWARDS A N N

BROTHERS,

A R B O R ,

M I C H I G A N

*942

INC.

INTRODUCTION The present monograph has been planned In such a way as to form a natural companion to the author’s volume Algebraic Topology appearing at the same time in the Col­ loquium Series and h£5?eafter referred to as AT. The topics dealt with have for common denominator the rela­ tions between polytopes and general topology. The first chapter takes up the relations between polytopes in gen­ eral and the topologies which they may receive and in these questions we lean particularly heavily upon J. Tukey. The second chapter completes in certain important points the treatment of singular elements of AT. The third chapter deals with mappings of spaces on polytopes and certain related imbedding questions; it contains also a modem treatment of retraction for separable metric spaces. The last chapter is devoted to the group of questions centering around the general concept of local connectedness. Comparisons with retracts are considered at length, there is a full treatment of the homology and fixed point properties. The chapter concludes with an outline of the relations with "homology" local connected­ ness (the so-called HLC properties). The general notations are those of AT. In addition to a short reference bibliography, a mere supplement to that of AT, there is also given a fairly comprehensive bibliography on locally connected spaces and retraction.

TABLE OP CONTENTS Page Chapter I . POLYTOPES................................................................ 11 §1. A ffin e Simplexes and C o m p le xe s.......................... 11 §2. Geometric C o m p le x e s ............................................... . . 9 §3. Comparison o f the T opologies A sso cia te d w ith A ffin e C o m p le x e s .......................................... . .16 16

.. ..

Chapter I I .

SINGULAR COMPLEXES.......................................... . .23 23

Chapter I I I . MAPPING AND IMBEDDING THEOREMS. RE­ TRACTION .................................................................... . .35 35 §1. Fundamental Mapping T h e o r e m ..............................35 . . 35 §2. A p p lic a tio n to Normal and Tychonoff Spacesa . 4 ^5 5 ces ^9 §3 . Compact Imbedding o f Separable M etric Spaces §^. T o p o lo g ica l Imbedding in E uclidean Spaces . 53 . . 58 § 5 . R e tra c tio n ................................................................58 Chapter IV. LOCAL CONNECTEDNESS AND RELATED TOPICS5 . 75 §1 . L o c a liz a tio n ................................................................75 . . 75 §2. P a r t ia l R e a liz a tio n o f Complexes. A p p licaattio n Ion . .81 81 to L ocal C o n n e cted n e ss.......................................... §3. R e latio n s Between the LC P ro p e rties and R e tra c tio n ............................................................... . .92 92 § 4 . C h a ra cte riza tio n o f the LC P ro p erties by Mappings o f Continuous C om p lexes..................... . .98 98 §5. Homology Theory o f LC Spaces .......................... ^ . .10104 §6 . Coincidences and F ixed P o i n t s .......................... . . 1112 12 §7. HLC Spaces. G eneralized M anifolds . . . . . . . 122 S p e cia l B i b l i o g r a p h y ............................................................... . .127127 General B ib lio g r a p h y .................................................... . •133 133 Index

. .

136

Chapter I. POLYTOPES §1 . AFFINE SIMPLEXES AND COMPLEXES 1. Affine Simplexes. In spite of the evident an­ alogy with the treatment of Euclidean simplexes of (AT, III, VIII). it will be more convenient and also clearer to repeat the necessary introductory definitions and proper­ ties. Our simplexes are considered here also as subsets of a real vector space ft whose elements are to be called points. (1 .1 ) DEFINITION. Let crp = aQ... ap be a p-simplex whose vertices are independent points of a real vector space 33 . By the af­ fine p-simplex associated with crp is meant the set, written cr^ given by (1.2) x =* x^a.^ (1.5)

p = o : x° = 1 ,

(1.4)

p > 0 : 0 < x i