3

PREFACE

The subject matte r of t h i s paper deals with the representatio n

problem for a collection of Prechet v a r i e t i e s known as Prechet surfaces (see

1.2)

1

. On the other hand, the representation problem may be stated , and a

p a r t i c u l a r form of a solution offered for al l Prechet v a r i e t i e s . This being

the case, the discussion will begin on general line s and the need for special-

izatio n will become apparent l a t e r , when i t will be observed that the present

frontier of mathematical knowledge and the general form for a desired solution

appears to dictate a specialization of the Prechet varietie s to be considered.

I t should be stated that an active effort i s made to do something

more than offe r a solutio n to the representatio n problem. In the f i r s t

chapter a measure of attention i s directed towards an indication of how the

pattern of research on thi s problem developed during the decades following the

i n i t i a l major assault on the problem by Kerékjártó [5]? This does not mean

t h a t the comments in question are devoted to tracin g the h i s t o r y of the

subject as such — for thi s and further bibliography the reader may consult

Youngs [17] — but rather that the emphasis i s on a discussion of the methods

of attack and the directions along which these methods evolved. In addition,

an attempt i s made to indicate those difficultie s which i n i t i a t e d the i n t r o -

duction of the topological tool s here employed. Consequently, i t i s hoped

that the f i r s t few paragraphs will serve two ends. For those who may wish to

learn about the problem but are uninterested in the detail s of the solution,

the discussion wil l serve, perhaps, as a means of acquiring a r e l a t i v e l y

painless understanding of the problem and a fair measure of intuitio n for the

subject . For those who wil l go on to study the proofs, i t i s hoped tha t

these paragraphs will provide an overall picture of the situation before one

i s plunged into details .

I t should be mentioned that active use will be made both of analytic

and algebraic topology; specifically , of the cyclic element theory and the

l . Numbers in parentheses refer to paragraphs in this paper.

a.' Numbers in brackets refer to the bibliography.