By Salzmann H.

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The global collapsing index of an (ma,mz,n)-modular quilt Q is defined to be the least common multiple of all of the collapsing indices of all vertices (of both types) and patches of Q. Similarly, the global collapsing index of an (ml, m2, oo)-modular quilt Q is defined to be the least common multiple of all of the collapsing indices of all vertices (of both types) of O. c c a @ d ~ g - e 6 Fig. 8. 9. 8 shows a (6, 5, (x~)-modular quilt Q, drawn using two conventions: First, we omit the dotted 1-seams of Q for clarity; and second, since Q happens to have genus 0, we draw Q by omitting a point "at infinity", or in other words, by thinking of Q as a planar diagram.

18. Replacing the Z-orbit representative (A,O) with the representative (A,j) = (A,O)ZJ has the effect of adding a boundary flow of +j to the dotted 1-ceUs around A in the associated quilt diagram. Proof. Clearly, only the dotted 1-cells touching A are affected. Therefore, suppose for r = 1, 2, the seam A is attached to the seam B at its Vr 1-cell and the seam C at its Vr--1 1-cell, as shown in Fig. 8. Then by Defn. 3 Quilt diagrams and quilts ', U. ' Fig. 8. Seams touching A (A, 0)V~ = (B, t), (C, 0)Vr = (A, u).

2. An oriented O-cell of X is defined to be a 0-cell of X together with a sign + or - . An oriented 1-ceU of X is defined to be a 1-cell e of X together with a choice of "direction of travel" along e (Fig. 3). More precisely, in the notation of Defn. 1, if Ia is the interval glued onto X ° to form e, an orientation on e is a choice of direction Cleft or right) on Ia. An oriented 2-cell of X is defined to be a 2-cell f of X together with a choice of "cyclic direction of travel" around f (Fig. 4).

### 16-dimensional compact projective planes with a collineation group of dimension >= 35 by Salzmann H.

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