Estimates for homological dimension of configuration spaces of graphs
Estimates for homological dimension of configuration spaces of graphs
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DOI：
10.4064/cm89-1-5
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发表时间：
2001
影响因子：
0.4
通讯作者：
J. Świa̧tkowski
中科院分区：
文献类型：
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作者：
J. Świa̧tkowski
我们证明了图Γ的一个配置空间的同调维度是由Γ中价大于2的顶点的个数b从上面估计的。我们证明了当n个Γ2b时，这个估计对于≥的n点配置空间是最优的。0。导言。设Γ是有限图，n是自然数。Γ的标记n点配置空间是Γ的第n次笛卡尔幂中的子空间CnΓ，由CnΓ：={(x1，.。。，xn)∈Γn：xi 6=xj，其中i 6=j}。考虑对称群Sn在空间CnΓ上的自然自由作用，空间Cn由σ(x1，.。。，xn)=(xσ(1)，.。。，xσ(N))，并放置CnΓ：=CnΓ/Sn.CnΓ空间称为Γ的(未标记)n点配置空间。本文报道了在理解图的配置空间的同调，甚至更一般的紧多面体的同调方面的部分进展。关于这一方向的另一个最新结果，见[G]。如果Γ的一个顶点v与至少三条边相邻，我们称它为分支的。我们用b=b(Γ)表示Γ中分支顶点的数目。本文的主要研究成果如下。0.1。定理。设Γ是有限图，n是自然数。(1)存在维度为Min(b(Γ)，n)的立方体复形KnΓ，它作为变形收缩嵌入到位形空间CnΓ中。(2)基本群π1(CnΓ)包含同构于自由交换群Z的子群，其中k=Min(b(Γ)，[n/2])，其中[x]表示x.2000数学学科分类：小学55M10；中学20J05，51F99的整数部分。提交人得到了波兰国家科学研究委员会(KBN)第2次拨款P03A 023 14的支持。
We show that the homological dimension of a configuration space of a graph Γ is estimated from above by the number b of vertices in Γ whose valence is greater than 2. We show that this estimate is optimal for the n-point configuration space of Γ if n ≥ 2b. 0. Introduction. Let Γ be a finite graph and n a natural number. The marked n-point configuration space of Γ is a subspace CnΓ in the nth cartesian power of Γ defined by CnΓ := {(x1, . . . , xn) ∈ Γ n : xi 6= xj for i 6= j}. Consider the natural free action of the symmetric group Sn on the space CnΓ defined by σ(x1, . . . , xn) = (xσ(1), . . . , xσ(n)) and put CnΓ := CnΓ/Sn. The space CnΓ is called the (unmarked) n-point configuration space of Γ . This paper reports on partial progress towards understanding the homology of configuration spaces of graphs, or even more generally of compact polyhedra. For another recent result in that direction, see [G]. We call a vertex v of Γ branched if it is adjacent to at least three edges. We denote by b = b(Γ ) the number of branched vertices in Γ . The main result of this paper is the following. 0.1. Theorem. Let Γ be a finite graph and n a natural number. (1) There exists a cube complex KnΓ of dimension min(b(Γ ), n) which embeds as a deformation retract into the configuration space CnΓ . (2) The fundamental group π1(CnΓ ) contains a subgroup isomorphic to the free abelian group Z with k = min(b(Γ ), [n/2]), where [x] denotes the integer part of x. 2000 Mathematics Subject Classification: Primary 55M10; Secondary 20J05, 51F99. The author was supported by the Polish State Committee for Scientific Research (KBN) grant 2 P03A 023 14.