10

THE REPRESENTATION PROBLEM

[I

essential to know [ V].

1.7 But to return to the main body of the argument, itisnot surprising

that thetwo obvious necessary conditions of1.5 lack sufficiency. For example,

suppose Ii - X - X is the set of points \2\ £ 1 on the complex ar-plane.

Let fx(z) ~ s and f^(s) - 2%. Then f±(X) = f^(X), but the mappings are

clearly not Frechet equivalent.

These necessary conditions are, of course, on a very easy level. A

muchmore sophisticated necessary condition, and the first non-trivial condition

to be known, was discovered by Kere'kja'rto* [ 5 ] in 1926 for Frechet sur-

faces of the type of the 2-sphere. (Kere'kjairto'1 s argument is readily gen-

eralized to cover all Frechet varieties, see Youngs [ 17 ]; in fact, Rado

[ 8 ] has recently used a new approach for this purpose which is the ulti-

mate in simplicity, and the reader should consult his paper on this point.)

Before stating the Kere'kjarto' condition it is well, from the

point of view of motivation, to look at the example of 1.4 showing that ex-

act matching is usually impossible. This example shows precisely where the

difficulty lies; it is because one of the mappings takes a non—degenerate con-

tinuum into a single point. Indeed, if f± ~ f' and neither / nor fQ

maps any non-degenerate continuum into a single point — such mappings are

called light (see 3.3) — then it can be shown that an exact matching is

possible, as is now well known and will be shown below. It was Kerekjarto'

who first observed this fortunate property for light mappings from 2-spheres.

He also used the fact that any mapping, f:X - » Y, can be factored and writ-

ten as the product, or composition, of two mappings, tfie first monotone (see

3.2) and the second light.

Specifically, if f:X — Y is a mapping, then there is a monotone

mapping %\X rj3 C and a light mapping l:X - Y such that / = Im; that is,

f(x) - l(m(x)) for every x ^ I (see the Eilenberg-Whyburn Factor Theorem,

3.5 ). (The letters m and / x will invariably be used for monotone map-

pings, I and k for light.) The space X is called the middle space of

the monotone light factorization, lm, of /. The situation can be repre-

sented in a commutative diagram: