Preface

Modern topology uses many different methods. In this book, we largely

investigate the methods of combinatorial topology and the methods of dif-

ferential topology; the former reduce studying topological spaces to investi-

gation of their partitions into elementary sets, such as simplices, or covers by

some simple sets, while the latter deal with smooth manifolds and smooth

maps. Many topological problems can be solved by using any of the two

approaches, combinatorial or differential; in such cases, we discuss both of

them.

Topology has its historical origins in the work of Riemann; Riemann's

investigation was continued by Betti and Poincaxe. While studying mul-

tivalued analytic functions of a complex variable, Riemann realized that,

rather than in the plane, multivalued functions should be considered on

two-dimensional surfaces on which they are single-valued. In these con-

siderations, two-dimensional surfaces arise by themselves and are defined

intrinsically, independently of their particular embeddings in

E3;

they are

obtained by gluing together overlapping plane domains. Then, Riemann

introduced the notion of what is known as a (multidimensional) manifold

(in the German literature, Riemann's term Mannigfaltigkeit is used). A

manifold of dimension n, or n-manifold, is obtained by gluing together over-

lapping domains of the space

Mn.

Later, it was recognized that to describe

continuous maps of manifolds, it suffices to know only the structure of the

open subsets of these manifolds. This was one of the most important rea-

sons for introducing the notion of topological space; this is a set endowed

with a topology, that is, a system of subsets (called open sets) with certain

properties.

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