By Mark Ronan
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17. A polarity of a generalized 1n-gon is an (ou ter) au tomorphism of order 2 interchanging the t\VO types of vertices. Show that the chanlbers fixed by a polarity are mutually opposite, and if there are no fixed chambers then every chamber is carried to an opposite one. 18. te quadrangle; 37 3. BUILDINGS (iii) for any line L and point p not on L, there is a unique point of L collinear with p. 19. , (s, t) = 0 Vs, t E S). Sho\v that this is a generalized quadrangle in the sense of Exercise 18.
There is an obvious morphislTI 0' : C ~ C C, I sending a strict hOlTIOtOpy class to the homotopy class containing it. Using 'if;: (C,c) Lly be replaced by one of type 1112. In particular if I is not red uced, the existence of a gallery of type I from x to y does not ilnply that 6(x, y) = 1'J, but on the other hand if 6(x, y) = rJ then there is a gallery of type f from x to y (Exercise 1) . 1). Here are SOUle elenlentary consequences of the definition: (0) ~ is connected, 6 maps onto W, and 6(x, y) 6(y, x)-l. (i) 6(x,y) = ri ~ x and yare distinct and i-adjacent. (ii) i- and j-adjacency are mutually exclusive for i f. j. (iii) If there is a gallery of type I (not necessarily reduced) from x to y, and if I is homotopic to g, then there is also a gallery of type 9 from x to y.