 
Summary: RESEARCH BLOG 9/20/04
OBERWOLFACH
Two weeks ago, I attended the conference in topology at Oberwol
fach. It was an odd mixture of people, since there were many homotopy
theorists, as well as some people studying Kleinian groups. Many of the
homotopy theorists seemed interested in the Novikov conjectures, and
therefore in K and L theory. A much stronger conjecture is the Borel
conjecture, which is that a manifold M which is a K(, 1) is homo
topy rigid (thanks to Kevin Whyte for clearing some of this up for me,
although take everything I mention with a grain of salt, since I don't
claim to really understand this). This means that if another manifold is
homotopy equivalent to M, then it is homeomorphic to M. There are
various generalizations of this conjecture to deal with groups with tor
sion, and spaces with nontrivial homotopy groups. This is known for
hyperbolic manifolds (Farrell and Jones in higher dimensions, Gabai,
Meyerhoff and N. Thurston in dimension 3), and I believe is known for
3manifolds satisfying the geometrization conjecture. An interesting
unknown special case of the Novikov conjecture is for mapping class
groups. It turns out that if one has a group acting on a compactifi
cation of a manifold, such that the action satisfies certain conditions,
