 
Summary: RESEARCH BLOG 6/9/04
EULER CHARACTERISTIC OF MANIFOLD K(, 1)S
Benson Farb mentioned an open question about manifolds recently.
Closed odddimensional manifolds have Euler characteristic 0. It is
conjectured that a closed, aspherical 2n manifold M has (1)n
(M)
0. This holds for hyperbolic manifolds, by the ChernGaussBonnet
theorem (and is probably known for homogeneous spaces in general),
but is not even known for nonpositively curved manifolds. I decided
to think about the first nontrivial special cases. It is certainly true in
dimension 2, so let's consider dimension 4. Equivalently, we'd like to
show that for M4
, if (M) < 0, then i(M) = 0, for some i = 2, 3, 4.
We need not consider 1(M) < , since in this case sign(M) =
sign( ~M), where ~M is the universal cover. For simply connected 4
manifolds, 1 = 3 = 0, by the Hurewicz theorem and PoincarŽe duality,
so certainly (M) > 0 in this case. Another obvious observation is
that the conjecture holds for bundles over lower dimensional manifolds.
Let's consider the simplest Morse functions. Let f : M4
R be a self
