Michael Joachim, Thomas Schick: Positive and negative results on the Gromov-Lawson-Rosenberg conjecture
The Gromov-Lawson-Rosenberg (GLR)-conjecture for a group $Gamma$
states that a closed spin manifold $M^n$ ($nge5$)
with fundamental group $ Gamma$ admits a
metric with $scal>0$ if and only if its $C^*$-index $alpha(M)in
KO_n(C^*_{red}(Gamma))$ vanishes.
We prove this for groups $Gamma$ with low-dimensional classifying space and
products of such groups with free abelian groups, provided the
assembly map for the group $Gamma$ is (split) injective (and $n$ large enough).
On the other hand, we construct a
$5$-dimensional spin
manifold $M$ which does not admit a metric with $scal>0$ but
has the property
that already the image of its
$KO$-orientation $pD[M]in KO_*(Bpi_1(M))$ vanishes.
Therefore a corresponding weakened version
of the GLR-conjecture is wrong.
Last we address non-orientable manifolds. We give a
reformulation of the minimal surface method of Schoen and Yau
(extended to dimension $8$) and introduce a
non-orientable (twisted) version of it. We then construct
a $5$-dimensional manifold whose orientation cover admits
a metric of positive scalar curvature and use the latter to
show that the manifold itself does not. The manifold also is
a counterexample to a twisted analog of the
GLR-conjecture because its twisted index vanishes.
MSC-number: 53C20 (global Riemannian Geometry)
Thomas Schick