On the Genus of Configuration Spaces.
Topology of Arrangements and Applications
October 07,2004 11:00 AM to 11:55 AM
Speakers:
Salvetti, Mario
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Summary: |
Let $W$ be a finite irreducible Coxeter reflection group acting on some
Euclidean space and let $M$ be the complement to the associated Coxeter
arrangement.
The focus of this lecture is on computing the Schwartz genus of the Galois
covering $M \to\ M/W$ onto the orbit space. If the arrangement is not of
type $A$, then the genus is the rank of the Coxeter group plus one (as
proven in [DeConcini-Salvetti, Math.Res.Lett., 2000]) while in case $A_n$
similar result was known to be true (see [Vassiliev, Transl. Math. Monog.,
98, 1992]) only when $n+1$ is a prime power. For $A_n$, we find an
explicit
obstruction class, belonging to $H_n(\Sigma_{n+1},H^n(P_{n+1},Z)), where
$\Sigma_k$ is the symmetric group on $k$ letters and $P_k$ is the pure
braid
group on $k$ strands, which vanishes precisely when the genus is lower
than
$n+1$. For $n+1=6$ we prove that such class vanishes, so the genus is
lower
than 6 (we also quote the recent progress by G. Arone). |
Keywords: |
Hyperplane Arrangements. |
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