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
. 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.
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