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Triangulations of tex2html_wrap_inline7800


In this section we briefly indicate how to construct a large family of examples of finite ordered sets that have a fixed point free automorphism and are such that all retracts have the fixed point property. This shows that despite some nice results for special classes of ordered sets the approach 3 in the introduction might not lead to a resolution of the fixed point problem, as there are too many forbidden retracts. Theorem 4.51 was revealed to the author by an anonymous referee. It also gives a negative answer to Problem 2 in [96]. The author would like to hereby express his gratitude to this referee.



Proof: Let tex2html_wrap_inline11248 be a retraction, i.e., a continuous idempotent map with tex2html_wrap_inline11250 . Then tex2html_wrap_inline11252 is isomorphic to a retract of the n-dimensional unit ball which has the topological fixed point property by Brouwer's fixed point theorem. \




Proof: Let tex2html_wrap_inline11286 be the truncated incidence lattice of the triangulation K of tex2html_wrap_inline7800 . Let tex2html_wrap_inline11292 be a nontrivial retraction with tex2html_wrap_inline11294 . Then there is a minimal element m of tex2html_wrap_inline11286 that is not in tex2html_wrap_inline11300 : Indeed otherwise r fixes all minimal elements of tex2html_wrap_inline11286 , which implies tex2html_wrap_inline11306 for all tex2html_wrap_inline11308 . Then there is a tex2html_wrap_inline11310 such that r(y)>y and r(p)=p for all tex2html_wrap_inline11316 . Then y has exactly one upper cover and there is a maximal element tex2html_wrap_inline11320 such that tex2html_wrap_inline11322 has exactly one point (otherwise y has more than one upper cover). Thus M is the only upper bound of y. But then every point x that is in the interior of the topological realization tex2html_wrap_inline11332 of y is such that for all tex2html_wrap_inline11336 small enough we have that tex2html_wrap_inline11338 is homeomorphic to the upper half space tex2html_wrap_inline11340 . Since tex2html_wrap_inline7800 is an n-dimensional manifold without boundary this is a contradiction to |K| being homeomorphic to tex2html_wrap_inline7800 .
Let tex2html_wrap_inline11350 be an order-preserving map. Let tex2html_wrap_inline11352 . For each minimal element tex2html_wrap_inline11354 choose a minimal element tex2html_wrap_inline11356 such that tex2html_wrap_inline11358 . For tex2html_wrap_inline11360 not minimal let


Then G is order-preserving on tex2html_wrap_inline11286 . Moreover tex2html_wrap_inline11368 is a retraction and tex2html_wrap_inline11368 induces a simplicial map on K that is a nontrivial retraction (we have shown above that the retract does not contain all minimal elements), which can be extended to a continuous retraction tex2html_wrap_inline11374 . Now by Lemma 4.47 tex2html_wrap_inline11376 has the topological fixed point property. Thus the continuous map on R[|K|] induced by G has a fixed point p. Let S be the smallest simplex in tex2html_wrap_inline11386 such that tex2html_wrap_inline11388 . Then G maps S to a sub-simplex of S, i.e., tex2html_wrap_inline11396 and thus G has a fixed point. Thus tex2html_wrap_inline11400 has a fixed point, which must be a fixed point of f. \


Proof: For each simplex S of K let A(S):=-S. By hypothesis this is well-defined. Since no simplex is equal to its antipode tex2html_wrap_inline11432 is a fixed point free order-preserving automorphism of tex2html_wrap_inline11286 . By Lemma 4.50 all nontrivial retracts of tex2html_wrap_inline11286 have the (order-theoretical) fixed point property. \


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Next: More Late-Breaking News ... Up: Order vs. Algebraic Topology Previous: Truncated Lattices