Higher Order Phantom Maps
A key problem of homotopy theory is to determine
if a given map f: X --> Y between two spaces is
homotopic to the constant map. This is a hard problem
which becomes much more difficult if the domain X
is infinite dimensional. One way to reduce the really
hard case to the merely hard case is to look for a
finite dimensional space K and a map g: K --> X such
that the composition K --> X --> Y is not trivial (i.e.,
homotopic to the constant map). If there is such a map
g then f cannot be trivial. However, it is possible that
there is no such map g and yet f is not the trivial map:
such a map f is called a phantom map.
Phantom maps are, by definition, hard to understand.
It is known that if there is one nontrivial phantom map
from X to Y, then there are uncountably many. Until
recently, however, there was no way to
distinguish any one of these maps from any other.
This talk will start from the definition of homotopy
(attention, graduate students!), wind its way through
some of the standard theory, and end with some new
ideas which enable us to distinguish phantom maps from
one another, and which show the existence of even