In this paper we present proofs of basic results, including those developed
so far by H. Bell, for the plane fixed point problem. Some of these results had
been announced much earlier by Bell but without accessible proofs. We define
the concept of the variation of a map on a simple closed curve and relate it to
the index of the map on that curve: Index = Variation + 1. We develop a prime
end theory through hyperbolic chords in maximal round balls contained in the
complement of a non-separating plane continuum $X$. We define the concept of an
{\em outchannel} for a fixed point free map which carries the boundary of $X$
minimally into itself and prove that such a map has a \emph{unique} outchannel,
and that outchannel must have variation $=-1$. We also extend Bell's linchpin
theorem for a foliation of a simply connected domain, by closed convex subsets,
to arbitrary domains in the sphere.
We introduce the notion of an oriented map of the plane. We show that the
perfect oriented maps of the plane coincide with confluent (that is composition
of monotone and open) perfect maps of the plane. We obtain a fixed point
theorem for positively oriented, perfect maps of the plane. This generalizes
results announced by Bell in 1982 (see also \cite{akis99}). It follows that if
$X$ is invariant under an oriented map $f$, then $f$ has a point of period at
most two in $X$.