Abstract. In 1969 R.H. Bing asked the following question: Is there a compact twodimensional polyhedron with the fixed point property which has even Euler characteristic? In this paper we prove that there are no spaces with these properties and abelian fundamental group. We also show that the fundamental group of an example cannot have trivial Schur multiplier.
Let G be a finite group and Ap(G) be the poset of nontrivial elementary abelian p-subgroups of G. Quillen conjectured that Op(G) is nontrivial if Ap(G) is contractible. We prove that Op(G) = 1 for any group G admitting a G-invariant acyclic p-subgroup complex of dimension 2. In particular, it follows that Quillen's conjecture holds for groups of p-rank 3. We also apply this result to establish Quillen's conjecture for some particular groups not considered in the seminal work of Aschbacher-Smith.
Abstract. We prove that there exists a compact two-dimensional polyhedron with the fixed point property and even Euler characteristic. This answers a question posed by R.H. Bing in 1969. We also settle another of Bing's questions.
IntroductionIn his influential article "The elusive fixed point property" [3], R.H. Bing stated twelve questions. Since then eight of these questions have been answered [6]. In this paper we answer Questions 1 and 8.Recall that a space X is said to have the fixed point property if every map f : X → X has a fixed point. Motivated by an example of W. Lopez What is the lowest dimension for a compact polyhedron X with the fixed point property and such that a space Y without the fixed point property can be obtained by attaching a disk D to X along an arc?The answer to this question is clearly greater than 1. A one-dimensional polyhedron X with the fixed point property is a tree, and then any space Y obtained by attaching a disk along an arc is a contractible polyhedron. According to C.L. Hagopian [6], Bing conjectured that the answer to Question 1.2 is 2. This is the content of Theorem 2.8.Acknowledgment: I am grateful to Jonathan Barmak, without his advice and suggestions this paper would not have been possible.
Bing groupsIf P is a presentation, the presentation complex of P will be denoted by X P . Presentation complexes are in fact polyhedra. If a finite group G is presented by a presentation P with g generators and r relators, then r − g is at least the number of invariant factors of H 2 (G). If this lower bound is attained for P, then the presentation is said to be efficient.2010 Mathematics Subject Classification. 55M20, 57M20, 57M05.
We prove that the simplicial complex whose simplices are the nonempty partial bases of Fn is homotopy equivalent to a wedge of (n−1)‐spheres. Moreover, we show that it is Cohen–Macaulay.
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