Thomas Schick: Integrality of L2-Betti numbers
- Autor: Thomas Schick
- Titel: Integrality of L2-Betti numbers (dvi) (pdf)
- Math. Ann. 317, 727-750 (2000), Erratum to appear
The Atiyah conjecture predicts that the $L^2$-Betti numbers of a finite
$CW$-complex with torsion-free fundamental group are integers.
We show that the Atiyah conjecture holds (with an additional
technical condition)
for direct and inverse limits of directed systems of groups for which it is true. As a
corollary it holds for residually torsion-free solvable groups,
e.g.~for pure braid groups or for
positive $1$-relator groups with torsion free abelianization.
Putting everything together we establish a new class of
groups for which the
Atiyah conjecture holds, which contains all free
groups and in particular is closed under taking subgroups, direct
sums,
free products,
extensions with
elementary amenable quotient,
and under direct and inverse limits of directed systems.
MSC: 55N25 (homology with local coefficients), 16S34 (group rings,
Laurent rings), 46L50
(non-commutative measure theory)
This is a corrected version of an older paper with the same title.
The proof of one of the basic results of the earlier version
contains a gap, as was kindly pointed out to me by Pere Ara.
This gap could not be fixed. Consequently, in this new version
everything based on this result had to be removed.
Thomas Schick