Topology of configuration space of two particles on a graph. II. (English) Zbl 1215.55006

This paper continues the investigation of the topology of the configuration space \(F(\Gamma ,2) := \Gamma\times\Gamma -\Delta\), where \(\Gamma\) is a finite graph and \(\Delta\) is the diagonal [K. Barnett and M. Farber, Algebr. Geom. Topol. 9, No. 1, 593–624 (2009; Zbl 1171.55007)]. A finite connected graph \(\Gamma\) which is not homeomorphic to the unit interval is called mature if the homomorphism \(H_1(F(\Gamma, 2))\rightarrow H_1(\Gamma\times\Gamma)\), induced from the subspace inclusion \(F(\Gamma, 2))\hookrightarrow \Gamma\times\Gamma\) is an isomorphism. According to results in [loc. cit.] no planar graph can be mature. In this paper maturity is established for many examples of graphs. This is done by proving a series of results describing the way the homology of \(F(\Gamma, 2)\) is affected by selectively changing \(\Gamma\). One main result of this paper (Theorem 1) shows that if \(\Gamma\) is mature and \(\hat\Gamma\) is obtained from \(\Gamma\) by adding an edge connecting two vertices \(u,v\) of \(\Gamma\), then if \(\Gamma -\{u,v\}\) is connected, \(\hat\Gamma\) is mature as well. As a corollary, it is shown that the complete graphs \(K_n\) for \(n\geq 5\) and the bipartite graphs \(K_{p,q}\) with \(p\geq 3\), \(q\geq 3\), are mature. The Betti numbers of the configuration spaces are computed in this case. It is pointed out that Theorem 1 of this paper contradicts Theorem 4.2 of C. W. Patty [Trans. Am. Math. Soc. 105, 314–321 (1962; Zbl 0113.16803)].
Various examples of non-mature graphs are described: (i) graphs with vertices of valence one, (ii) graphs such that removing the closure of an edge makes it disconnected, (iii) graphs with a double edge, and (iv) graphs that are wedges or double wedges of two connected subgraphs \(\Gamma_1\) and \(\Gamma_2\) both of which are connected and not homeomorphic to the unit interval. On the other hand, mature graphs can be produced in the following way: if \(\Gamma = \Gamma_1\cup\Gamma_2\) is the union of two mature subgraphs such that the edges incident to any vertex \(v\in \Gamma_1\cap\Gamma_2\) lie either all in \(\Gamma_1\) or all in \(\Gamma_2\), and if \(\Gamma_1\cap\Gamma_2\) is connected, then \(\Gamma\) is mature.
Techniques of proof rely on the intersection form \(I_\Gamma\) introduced in [Barnett and Farber, op. cit.] and the linking homomorphism introduced in §4 of this paper.
The paper ends by making the conjecture that a connected nonplanar graph is mature if and only if it admits a decomposition as a wedge or double wedge. Also it is indicated that no examples of graphs \(\Gamma\) are known such that \(H_1(F(\Gamma, 2))\) has nontrivial torsion.


55R80 Discriminantal varieties and configuration spaces in algebraic topology
57M15 Relations of low-dimensional topology with graph theory
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