Foreword

Topology studie s th e propertie s o f geometrica l object s tha t remai n un -

changed under transformations calle d homeomorphisms an d deformations.

The initial aquaintance with this field is hindered by the fact that rigorous

definitions o f eve n th e simples t notion s o f topolog y ar e rather abstrac t o r

very technical. Fo r this reason the first really meaningful (an d in fact readily

understandable) topological theorems appear only after tedious preliminaries

have been overcome. Thi s preliminary work is mostly devoted to the detailed

and accurate proofs of intuitively obvious statements: admittedl y not a very

exciting activity.

This boo k i s a n introductor y cours e i n topolog y o f rathe r untraditiona l

structure. W e begin by defining the main notions in a tangible and perceptible

way, o n an everyday level, an d as we go along we progressively mak e them

more precise and rigorous, reaching the level o f fairly sophisticate d proofs .

This allows us to tackle meaningful problems from the very outset with some

success.

Another unusual trait of this book is that it deals mainly with constructions

and maps (of surfaces, knots, and links in space), rather than with proofs of

general theorems implyin g tha t certai n map s and constructions don' t exist .

Such proofs, usuall y based on complicated invariant s (e.g. , so-called homo-

topy and homology functors), are in fact a more traditional activity for topol-

ogists, but are not the main subject matter of this book. W e do consider some

invariants, but only simple and effective ones.

The (numerous) illustrations are essential. I n many parts of the book they

are more important than the text, which is then little more than a commentary

to the pictures.

In the study of mathematics, problem solving plays a crucial role. Readin g

ready-made proofs of theorems is a poor substitute for trying to prove them

on your own. Man y statement s tha t th e reade r ca n profitabl y thin k abou t

himself appea r i n th e for m o f problems . Thes e problem s ar e an inheren t

part of our exposition, and therefore their solutions are presented at the end

of each section.

A bibliography, mainl y consistin g o f book s tha t w e recommen d fo r th e

vii