Summary: RESEARCH BLOG 6/9/04
EULER CHARACTERISTIC OF MANIFOLD K(, 1)S
Benson Farb mentioned an open question about manifolds recently.
Closed odd-dimensional manifolds have Euler characteristic 0. It is
conjectured that a closed, aspherical 2n manifold M has (-1)n
0. This holds for hyperbolic manifolds, by the Chern-Gauss-Bonnet
theorem (and is probably known for homogeneous spaces in general),
but is not even known for non-positively curved manifolds. I decided
to think about the first non-trivial 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-