[Binghamton Geometry/Topology Seminar] October 13, joint with Combinatorics Seminar

[Marco Varisco]

Binghamton Geometry/Topology Seminar, joint with Combinatorics Seminar
http://www.math.binghamton.edu/dept/topsem/

Date: Tuesday, October 13, 2009

Time: 1:15 pm

Place: Library North 2205

Speaker: Max Wakefield (United States Naval Academy, Annapolis)

Title: Topological formality of arrangements of subspaces derived from
edge-colored hypergraphs

Abstract: An arrangement of linear hyperplanes in C^n generates
subspaces defined by intersections of hyperplanes. A choice of such
subspaces is an arrangement of subspaces. When the subspaces are
hyperplanes the arrangement's characteristic polynomial carries
information about the topology of M, the complement of the subspaces
in C^n; this is a fundamental theorem of Orlik and Solomon.
An arrangement of subspaces of the braid arrangement (the arrangement
that consists of all hyperplanes with equations x_i=x_j) can be
encoded by an edge-colored hypergraph. The characteristic polynomial
of this type of subspace arrangement is given by a generalized
chromatic polynomial of the associated edge-colored hypergraph.
However, this polynomial is less informative than in the case of
hyperplane arrangements. Stronger topological information about M can
be found directly in the hypergraph.
A Massey product is an algebraic simplification in cohomology. I will
present a sufficient condition for the existence of non-trivial Massey
products in the cohomology of M. The condition is proved by studying a
spectral sequence associated to the Lie coalgebras of Sinha and
Walter. These coalgebras are constructed from the cohomology of M.
If time permits I will construct a family of subspace arrangements
whose intersection lattices have the shape of Pascal's triangle. Even
though the intersection lattices are not geometric, the complex
complements of the arrangements have the property of rational
formality, i.e., their homotopy type is determined by their rational
cohomology.
Everything I will talk about is combinatorial. Some of the topics are
in combinatorial topology, but I will try my best to not be overly
technical.