[Binghamton Geometry/Topology Seminar] Geometry/Topology and Combinatorics seminar Tuesday

[zaslav at math.binghamton.edu]

The Combinatorics seminar tomorrow is likely to be of interest to
topologists, so I'm sending a special reminder to "you guys".
Tom Zaslavsky

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Tuesday, September 9
Speaker: Emanuele Delucchi (Binghamton)
Title: Finite Reflection Groups, Non-Crossing Partitions, and a
Theorem of Deligne
Time: 1:15 - 2:15
Room: LN-2205

Consider an arrangement A (a finite set) of real hyperplanes that
dissect real space into regions that are simplicial cones. Let M be
the complex complement of the complexification of A. In 1972 Deligne
proved that M is an Eilenberg-Maclane space. The proof focuses on U,
the universal covering space of M, and proceeds by first showing that
simpliciality of the regions is equivalent to the existence of a
certain normal form for the morphisms of the fundamental groupoid of
M. This, together with the fact that U can be constructed directly
from the groupoid, implies contractibility of U.

I will sketch Deligne's proof and show the connection to a similar
argument of David Bessis in 2005 that proves asphericity of M for the
arrangements of reflecting hyperplanes of finite unitary reflection
groups. The groupoids involved in Bessis' work have structure based on
the combinatorics of "non-crossing partitions". The geometric
significance of this is not yet clear.

I will give background and motivation for future talks in the
combinatorics seminar about non-crossing partitions and
generalizations. A better understanding of their combinatorial
structure could give us insight into the topology of complex
hyperplane arrangements!


http://www.math.binghamton.edu/dept/ComboSem/abstract.200809del.html