[UC Top] fixed point theory

[Kathleen Ponto]

Let f be a self map of a space X.  A point x in X is a fixed point of f if
f(x)=x.  The Lefschetz fixed point theorem gives an ("easy to compute")
invariant that indicates when a self map of a polyhedron has at least one
fixed point.  This is really a consequence of the following statement of 
the theorem:

Theorem: Let f be a self map of a polyhedron X, then the trace of the
induced map on rational cohomology is equal to the index of f.

The index is a function which assigns to each self map a rational number. 
It can be described many ways and it is a keeps track of the local 
behavior of the map near the fixed point.

While this theorem can sometimes indicate if a map has a fixed point, it
does not give any indication of how many fixed points the map has.  There 
is an alternative invariant that uses information about the fundamental 
group to give an lower bound for the number of fixed points of the map. 
This invariant, called the Nielsen number, is defined by partitioning the 
fixed points into classes and then counting the number of essential 
classes.  A class is essential if the index is nonzero.  The Nielsen 
number, and the index, are invariant under homotopy and this gives a lower 
bound on the number of fixed points any map homotopic to the original map.

The Nielsen number has its own problems.  The bound it produces is not
always sharp.  It is known to be sharp for all maps of some spaces and 
there are also examples where it is not.  The Nielsen number is also not 
easy to compute.  There are some situations where computing the Nielsen 
number reduces to computing the number of fixed point classes.