The "sticky conjecture" states that a geometric lattice is modular if and only if .azy two of its extensions can be "glued together". It is known to be true as far as rank 3 geometries are corrcerned. In this paper we show that it is sufficient to consider a very restricted class of rank 4 geometries in order to settle the question. As a corollary we get a characterization of uniform sticky matroids, which has been found by Poljak and Tutzik in i9iib.
Let L be a geometric lattice and denote the set of copoints of L by %'. 3 c ,947 is called a linear subclass, if for every coline c of L, either c is covered by at most one element of 2 or all copoints covering c are contained in 5!! (cf. ES]). The set of all linear subclasses of L can be ordered by inclusion, giving rise to a lattice with point set equal to X This is called the extension-lattice of L and is denoted by E(L) (cf. [2, 31).
An important question arising from this, is to investigate, how far the geometric lattice L is determined by its extension lattice E(L)' For example, one could ask: Given two geometric lattices L and L' having isomo_rphic extension lattices, must L and L' be isomorphic. In general, the answer is: e.g., any projective geometry L of rank at least 4. The removal of a point will not change the incidence relation between t copoints and colines of L, and the extension lattice will not be changed.
owever, lf L is of rank 3, the sit becomes different. Mere it is easy to see (cf. Section 2), that L is modular if and only if E(L) is. As we v&l show in Section 3, this observ ion is related to another characterization of modular rank 3 geometries, found by Poljak and 'Iurzik (cf.
finite rank 3 geometry is odular if and only if it is 01 rted by the esearch Associatio