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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 [6] and by T. McKee and E. Prisner in [80]. 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 [80] from their original version for modulo 2 homology to integer homology. It becomes necessary, since the beautiful geometric approach to homology in [80] seems to depend on the fact that -1=1 in tex2html_wrap_inline11440 , 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 tex2html_wrap_inline11478 be a (q-2)-dimensional oriented simplex that occurs in one of the (q-1)-dimensional simplexes tex2html_wrap_inline11484 , with tex2html_wrap_inline11486 . Then tex2html_wrap_inline11488 occurs once (positive or negative) in tex2html_wrap_inline11490 , but by assumption it does not occur in tex2html_wrap_inline11492 Therefore tex2html_wrap_inline11488 occurs in an even number, say 2k, of the tex2html_wrap_inline11498 ( tex2html_wrap_inline11486 ) and it occurs such that in k of the tex2html_wrap_inline11490 the summand tex2html_wrap_inline11488 is positive and in the other k boundaries it is negative. If tex2html_wrap_inline11488 is a positive summand of tex2html_wrap_inline11490 , then tex2html_wrap_inline11478 is a negative summand of tex2html_wrap_inline11516 and vice versa. Thus no multiple of tex2html_wrap_inline11478 occurs in tex2html_wrap_inline11520 by the observed cancellation. The conclusion follows. \


Proof: Assume tex2html_wrap_inline11540 , tex2html_wrap_inline11542 . If q=1 assume tex2html_wrap_inline11546 , if tex2html_wrap_inline10554 assume tex2html_wrap_inline11550 .
First, we show that every homology class of q-dimensional cycles has a representative that does not contain v. Assume tex2html_wrap_inline10554 . Let tex2html_wrap_inline11558 be a representative of the homology class tex2html_wrap_inline11560 , with tex2html_wrap_inline11562 . If t=0, we are done, so we can assume tex2html_wrap_inline11566 . By Lemma 4.53, tex2html_wrap_inline11568 is a (q-1)-cycle of G and hence also of tex2html_wrap_inline11574 . Since tex2html_wrap_inline11550 there are q-simplexes tex2html_wrap_inline11580 , tex2html_wrap_inline11582 , tex2html_wrap_inline11584 of tex2html_wrap_inline11574 such that tex2html_wrap_inline11588 But then




Thus tex2html_wrap_inline11592 forms a q-dimensional cycle with vertices in tex2html_wrap_inline11596 . Finally C and C' are homologous, since


In case q=1, let tex2html_wrap_inline11604 be a representative of the homology class tex2html_wrap_inline11560 , with tex2html_wrap_inline11562 . t must be even, say t=2k and k of the tex2html_wrap_inline8526 must be 0, the rest 1 (otherwise the boundary of C contains a multiple of [v]). Since tex2html_wrap_inline11546 , we have by Lemma 4.19 that tex2html_wrap_inline11574 is connected. However then tex2html_wrap_inline11630 , 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 tex2html_wrap_inline10554 .
If we can show now that for any q-cycles C,C' that do not contain v we have tex2html_wrap_inline11642 iff tex2html_wrap_inline11644 then the map that maps every homology class tex2html_wrap_inline9130 in tex2html_wrap_inline11648 to the corresponding homology class in tex2html_wrap_inline11650 defined by the representatives of tex2html_wrap_inline9130 that do not contain v is an isomorphism and we are done. To do so first note that the direction `` tex2html_wrap_inline8148 " is trivial. For the other direction let tex2html_wrap_inline11658 with tex2html_wrap_inline11660 with tex2html_wrap_inline11562 . Then


with tex2html_wrap_inline11666 . By Lemma 4.53 tex2html_wrap_inline11668 is a q-cycle and since tex2html_wrap_inline11542 there must be (q+1)-dimensional simplexes in tex2html_wrap_inline11574 such that


But then


with the cancellation in the last step happening since tex2html_wrap_inline11668 is a cycle in tex2html_wrap_inline11574 . 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. [6]) 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 [29] we refer the reader to the original.

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