Minimal CWComplexes for Complements of Reflection Arrangements of Type A_(n1) and B_(n)
An arrangement of hyperplanes (or just an arrangement) A is a finite collection of linear subspaces of codimension 1 in a finite dimensional vector space. Each hyperplane H is the kernel of a linear function αH, which is unique up to a constant. ARn−1 denotes the braid arrangement in Rn, consist...
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Format:  Doctoral Thesis 
Language:  English 
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PhilippsUniversität Marburg
2009

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Summary:  An arrangement of hyperplanes (or just an arrangement) A is a finite collection
of linear subspaces of codimension 1 in a finite dimensional vector space.
Each hyperplane H is the kernel of a linear function αH, which is unique up
to a constant.
ARn−1 denotes the braid arrangement in Rn, consisting of the hyperplanes
Hi,j := {x ∈ Rn  xi = xj}, for 1 ≤ i &amp;amp;amp;lt; j ≤ n.
BR n denotes the arrangement in Rn which in addition to the hyperplanes Hi,j
of the braid arrangement consists of the hyperplanes Hi,−j := {x ∈ Rn  xi = −xj}, for 1 ≤ i &amp;amp;amp;lt; j ≤ n and the coordinatehyperplanes
Hi := {x ∈ Rn  xi = 0}, for i = 1, . . . , n.
A complexification of a real hyperplane arrangement in Rn is defined to
be the hyperplane arrangement in Cn which is defined by the same linear
forms.
We omit the index C and denote by An−1 and Bn the complexifications of
the real arrangements AR n−1 and BR n, respectively. The notation is chosen
according to the respective reflection groups of type An−1 and Bn.
For an arrangement of hyperplanes A we denote by M(A) the complement
of the union of all hyperplanes of A. The complements M(An−1) and M(Bn)
of the complexifications of the two arrangements above are the objects of our
study.
The topology of such complements have been the subject of studies since the
early 1970’s. The development started in 1972, when P. Deligne proved that
the complement of a complexified arrangement is K(π, 1) when the chambers
of the subdivision of Rn induced by the hyperplanes are simplicial cones [7].
1 With regard to this thesis one result of M. Salvetti from 1987 is of great importance.
He proved that the complement of a complexified real hyperplane arrangement is homotopy equivalent to a regular CWcomplex [18].
Since the groups Hi(Xi,Xi−1) of the cellular cochain complex of a CWcomplex X are free abelian with basis in onetoone correspondence with the icells of X, we call a CWcomplex minimal if its number of cells of dimension i equals the rank of the cohomology group Hi(X,Q).
Taking the regular CWcomplexes, which are based on Salvetti’s work, as a starting point, we derive minimal CWcomplexes An−1 and Bn for the complements M(An−1) ⊂ Cn and M(Bn) ⊂ Cn of the complexifications of the two arrangements above. Hence, we deduce CWcomplexes which are homotopy equivalent to M(An−1) or M(Bn) and which have a minimal number of cells.
In order to decrease the number of cells, discrete Morse Theory provides our
basis tool. It was developed by R. Forman in the late 1990’s. Discrete Morse
Theory allows to decimate the number of cells of a regular CWcomplex without
changing its homotopy type.
Parallel to our work, a general approach to finding a CWcomplex homotopic
to the complement of an arrangement using discrete Morse theory was developed in [19]. Our approach is different for the cases studied and leads to a much more explicit description than the statement in [19].
It is well known that the rank of the cohomology groups Hi(M(An−1),Q) and Hi(M(Bn),Q) of the complementsM(An−1) andM(Bn) equals the number of elements of length i in the underlying reflection groups Sn and SB n , respectively [1]. Here, Sn is the symmetric group and SB n is the group of signed permutations, consisting of all bijections ω of the set [±n] := {1, . . . , n,−n, . . . ,−1} onto itself, such that ω(−a) = −ω(a) for all a ∈ [±n].
Indeed, the numbers of cells of the minimal complexes An−1 and Bn are equal to the numbers of elements in Sn and SB n , respectively.
The cellorder of a CWcomplex X is defined to be the order relation on the
cells of X with σ ≤ τ for two cells σ, τ of X if and only if the closure of σ is
contained in the closure of τ . The poset of all cells of X ordered in this way
is called the face poset of X.
A main part of this thesis is devoted to the cellorders of the minimal CWcomplexes. In case of the complex An−1 the face poset turns out to have a concise description.
The combinatorics of the face poset of Bn seems to be too complicated to be
described through a concise and explicit rule. Thus we formulate a description
in terms of mechanisms which allow to construct the cells B with A &amp;amp;amp;lt; B
from a given cell A. Even though this description is relatively compact, there
2 is still a lot of combinatorics included that has yet to be discovered. 

DOI:  https://doi.org/10.17192/z2009.0111 