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In May-June 1996 results that give algebraic, resp. combinatorial proofs   for the result that every finite truncated noncomplemented lattice has the fixed point property were announced by K. Baclawski in  and by T. McKee and E. Prisner in . We will present McKee and Prisner's approach here as their methods are natural applications of what is discussed in this chapter. The following is thus nothing but a translation of the results in  from their original version for modulo 2 homology to integer homology. It becomes necessary, since the beautiful geometric approach to homology in  seems to depend on the fact that -1=1 in , while the arguments generalize beyond this visualization. One just needs to carefully keep track of the signs at times. Proof: In case q=1, there is nothing to prove (the boundary of a sum of zero-dimensional simplexes is 0). Let be a (q-2)-dimensional oriented simplex that occurs in one of the (q-1)-dimensional simplexes , with . Then occurs once (positive or negative) in , but by assumption it does not occur in Therefore occurs in an even number, say 2k, of the ( ) and it occurs such that in k of the the summand is positive and in the other k boundaries it is negative. If is a positive summand of , then is a negative summand of and vice versa. Thus no multiple of occurs in by the observed cancellation. The conclusion follows. \ Proof: Assume , . If q=1 assume , if assume .
First, we show that every homology class of q-dimensional cycles has a representative that does not contain v. Assume . Let be a representative of the homology class , with . If t=0, we are done, so we can assume . By Lemma 4.53, is a (q-1)-cycle of G and hence also of . Since there are q-simplexes , , of such that But then Thus Thus forms a q-dimensional cycle with vertices in . Finally C and C' are homologous, since In case q=1, let be a representative of the homology class , with . t must be even, say t=2k and k of the must be 0, the rest 1 (otherwise the boundary of C contains a multiple of [v]). Since , we have by Lemma 4.19 that is connected. However then , since in a connected graph any sum of differences of zero-dimensional simplexes is the boundary of a sum of 1-dimensional simplexes. Now the argument continues as in the case .
If we can show now that for any q-cycles C,C' that do not contain v we have iff then the map that maps every homology class in to the corresponding homology class in defined by the representatives of that do not contain v is an isomorphism and we are done. To do so first note that the direction `` " is trivial. For the other direction let with with . Then with . By Lemma 4.53 is a q-cycle and since there must be (q+1)-dimensional simplexes in such that But then with the cancellation in the last step happening since is a cycle in . This finishes the proof. \ This together with the removal of escamotable points as discussed in section 4.5 gives a proof of Baclawski and Björner's result that is entirely algebraic. (Contractibility of the neighborhood was needed to guarantee removability of the point without affecting the homology. The contractibility of escamotable graphs was a nice consequence, which is however stronger than what was needed. A notion ``vertex is weakly escamotable iff its pointed neighborhood is acyclic" is strong enough for our purposes.)
Baclawski's argument (cf. ) proves a result similar to Corollary 4.55 about structures called ``pseudo cones" using neither algebra, nor topology. His fixed point algorithm for pseudo cones does not use any previously known method and as with  we refer the reader to the original.     Next: Applications/New Directions Up: Order vs. Algebraic Topology Previous: Triangulations of

Bernd.S.W.Schroder