Sam Smith, Saint Joseph's University, Philadelphia

Title: Actions of spaces of self-equivalences on the components of a
function space

This is joint work with Greg Lupton.  We give a general method  for
showing two components of a function space map(X, Y)  have the same
homotopy type.  We describe certain group-like actions on  map(X,Y).   Our
basic results  assert that if maps f, g : X -> Y are in the same orbit
under such an action, then the components map(X, Y;f) and map(X, Y;g)
have the same homotopy type.  Applying our method to the natural
group-like action of the space of self-equivalences of  Y on map(X, Y)
we obtain an extension of an old result of G. Whitehead on the
fibre homotopy types of evaluation fibrations.