Minimal CW-complexes for complements to line arrangements via Discrete Morse Theory

Let A be an arrangement of hyperplanes in CN and let M(A) be its complement. From [DP03], [Ra02] it follows that M(A) has the homotopy type of a minimal CW-complex, i.e. a complex having as many kcells as the k th Betti number (k 0) (see also [PS02]). Such result appears as an \existence-like" theorem, and a description of a minimal complex, in the real dened case, was found in [SS07] in terms of an explicit construction of a discrete vector eld over a well-known CWcomplex (see [Sal87]). A rst partial description of the minimal complex was done in [Yo05] (see also [D08] for a combinatorial construction, and related works by [DS07] and [MS09] for a generalization to certain classes of subspace arrangements). Discrete Morse Theory (see [Fo98], [Fo02]) yields several informations about the minimal complex, including an algebraic Morse complex for cohomology (see [SS07] and [GS09]), and a constructive procedure to understand the attaching maps of the cells. In general, one obtains minimal number of cells but with \redundant" attaching maps. What may happen is that k1-cells can appear in the attaching map of a k-cell ek many more times than what is necessary: there are several pairs of the same cell which can be homotopically deleted. So, a natural problem is: starting from the above construction, produce a minimal complex with \reduced" attaching maps. In this paper we solve this problem in the two-dimensional case (see theorem 2).