Stefam Jackowski: Homotopy limits in topology and geometry

Abstract: Homotopy colimits of diagrams in variuos categories, a
"thick" and homotopy invariant version of direct limit, turned out to
be an extremly useful tool in homotopy theory. Cohomology of a space
decomposed into a homotopy colimit of a diagram can be calculated
using the Bousfield-Kan spectral sequence, whose E_2-term can be
expressed in terms of derived functors of the inverse limit over the
indexing category of the diagram.  The classical obstruction theory
for extending maps on subsequent skeleta of CW-complexes generalizes
to spaces decomposed into homotopy colimits - obstructions are again
elements of the derived functors.  Following work of D.Notbohm,
T.Panov, N.Ray and R.Vogt, I'll describe how homotopy colimits provide
a unifying thread for results on classifying spaces of compact Lie
groups, toric manifolds and their homotopy orbit spaces.