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.In this note, we show how to overcome this difficulty by using the sum
of the 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 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)
