Rational model of subspace complement on atomic complex

Sergey Yuzvinsky

Abstract: In [12], a rational model of the complement $C(X)$ to a complex subspace arrangement $X$ is constructed that uses as the underlying complex the direct sum of the order (flag) complexes for all the intervals $[0,A]$ of the intersection lattice $L(X)$ of $X$. The product in this model is defined via a messy algorithm involving the shuffle product of flags (see section 3). The atomic complex of a lattice is typically much smaller than its order complex although homotopy equivalent to the latter. Thus the sum of atomic complexes of the intervals of $L(X)$ could be used potentially for computation of the algebra $H^*(C(X),\Bbb Q)$ instead of the order complexes. However there is no natural multiplication on this sum that induces the right multiplication on homology. \par In this note, we show how to overcome this difficulty by using the sum of the {\it relative} atomic complexes. The relative atomic complex of an interval $[0,A]$ can be interpreted as the factor complex of the simplex on all the atoms under $A$ over the atomic complex of $[0,A]$. On the sum of these complexes, the needed multiplication is given by a very simple and natural formula. Roughly speaking, the product of two sets of atoms is either 0 or the union of these sets up to a sign. In cases where generators of local homology of $L(X)$ can be given explicitly one can use this formula and try to get a presentation of the ring $H^*(C(X),\Bbb Q)$. We briefly consider the simplest case of so called {\it homologically geometric} lattices that covers geometric lattices and lattices of types $\Pi_{n,k}$. The most important unsolved problems (see section 4) include the question of naturality and (related) question of integer coefficients. I include a conjecture about the latter problem (Conjecture 3.3) that was proved recently in two particular cases in [7] and [6]. (From the introducrtion)

Classification (MSC2000): 52B30; 05B35

Full text of the article:

• Compressed PostScript file (63 kilobytes)
• PDF file (173 kilobytes)