Minimal CW-Complexes for Complements of Reflection Arrangements of Type A_(n-1) 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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1. Verfasser: Djawadi, Daniel
Beteiligte: Welker, Volkmar (Prof.) (BetreuerIn (Doktorarbeit))
Format: Dissertation
Sprache:Englisch
Veröffentlicht: Philipps-Universität Marburg 2009
Mathematik und Informatik
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Zusammenfassung: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 < 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 < 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 CW-complex [18]. Since the groups Hi(Xi,Xi−1) of the cellular cochain complex of a CW-complex X are free abelian with basis in one-to-one correspondence with the i-cells of X, we call a CW-complex minimal if its number of cells of dimension i equals the rank of the cohomology group Hi(X,Q). Taking the regular CW-complexes, which are based on Salvetti’s work, as a starting point, we derive minimal CW-complexes 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 CW-complexes 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 CW-complex without changing its homotopy type. Parallel to our work, a general approach to finding a CW-complex 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 cell-order of a CW-complex 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 cell-orders of the minimal CW-complexes. 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 < 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