Small non-Leighton two-complexes. (English) Zbl 1529.57007
Leighton’s Theorem [F. T. Leighton, J. Comb. Theory, Ser. B 33, 231–238 (1982; Zbl 0488.05033)] says:
If two graphs have a common covering, then they have a common finite sheeted covering.
It is reasonable to ask if there are higher dimensional versions of this result, but the torus \(S^1\times S^1\) and the orientable closed surface \(S_2\) of genus 2 show that care must be exercised: both have universal covers homeomorphic to the plane \(\mathbb{R}^2\), yet they have no common finite sheeted cover. Consequently, conditions must be imposed in order to have a tractable problem in higher dimensions.
One way to do this is to recognize that a graph, along with its topological structure, also possesses a combinatorial structure. Thus, imposing combinatorial conditions on the higher dimensional spaces and the maps involved is reasonable. See, for example, Abello, Fellows, and Stillwell [J. Abello et al., Australas. J. Comb. 4, 103–112 (1991; Zbl 0763.05035)] and T. W. Tucker [Lect. Notes Math. 1440, 192–207 (1990; Zbl 0713.57001)].
Then, an interesting version of this problem for higher dimensional combinatorial objects is: Is it true that for finite CW complexes \(K_1\) and \(K_2\), if there exist a CW complex \(K\) and cellular covering projections \(f_1:K \rightarrow K_1\) and \(f_2:K \rightarrow K_2\), then there must be a finite CW complex with the same property?
Note that it suffices to consider the problem for 2-dimensional CW complexes. For, if \((K_1,K_2)\) are a pair of finite \(n\)-dimensional CW complexes, \(n \geq 2\), which have a finite common cellular cover \(K\) with cellular covering projections \(f_i:K\rightarrow K_i\), \(i=1,2\) , then the 2-skeletons of the \(K_i\) are cellularly covered by the restrictions of the \(f_i\) to the necessarily finite 2-skeleton of \(K\).
Wise answers the above question in the negative. He showed [D. T. Wise, Comment. Math. Helv. 82, No. 4, 683–724 (2007; Zbl 1142.20025)] that there is a pair \((K_1,K_2)\) of finite 2-dimensional CW complexes which have a common infinite cellular cover \(K\), but which have no finite CW complex which cellularly covers both. Call such a pair a non-Leighton pair.
Wise’s non-Leighton pairs contain as few as six 2-cells. The challenge then becomes that of determining how “small” 2-dimensional cellular non-Leighton pairs can be. That is, what is the minimum number of cells, or more specifically, the minimum number of 2-cells, possible in a 2-dimensional non-Leighton CW complex pair.
The authors of the manuscript under review prove
Theorem. The standard 2-dimensional CW complexes associated to the presentations \[ H_{\epsilon} = \langle a,c,d: a^{[c^d,c]}=a^{\epsilon},(c^3)^d=c^5 \rangle , \] \(\epsilon = \pm 1\), containing one vertex, three 1-cells, and two 2-cells, are a non-Leighton pair. That is, they have a common cellular covering, but no common finite cellular covering.
The \(H_{\epsilon}\) are amalgamated products of the fundamental group of the torus \(BS(1,1)\) and the Baumslag-Solitar group \(BS(3,5)\), amalgamated over a cyclic group.
The question of whether there exist non-Leighton pairs of 2-dimensional CW complexes with only one 2-cell is, at this time, still open.
It is reasonable to ask if there are higher dimensional versions of this result, but the torus \(S^1\times S^1\) and the orientable closed surface \(S_2\) of genus 2 show that care must be exercised: both have universal covers homeomorphic to the plane \(\mathbb{R}^2\), yet they have no common finite sheeted cover. Consequently, conditions must be imposed in order to have a tractable problem in higher dimensions.
One way to do this is to recognize that a graph, along with its topological structure, also possesses a combinatorial structure. Thus, imposing combinatorial conditions on the higher dimensional spaces and the maps involved is reasonable. See, for example, Abello, Fellows, and Stillwell [J. Abello et al., Australas. J. Comb. 4, 103–112 (1991; Zbl 0763.05035)] and T. W. Tucker [Lect. Notes Math. 1440, 192–207 (1990; Zbl 0713.57001)].
Then, an interesting version of this problem for higher dimensional combinatorial objects is: Is it true that for finite CW complexes \(K_1\) and \(K_2\), if there exist a CW complex \(K\) and cellular covering projections \(f_1:K \rightarrow K_1\) and \(f_2:K \rightarrow K_2\), then there must be a finite CW complex with the same property?
Note that it suffices to consider the problem for 2-dimensional CW complexes. For, if \((K_1,K_2)\) are a pair of finite \(n\)-dimensional CW complexes, \(n \geq 2\), which have a finite common cellular cover \(K\) with cellular covering projections \(f_i:K\rightarrow K_i\), \(i=1,2\) , then the 2-skeletons of the \(K_i\) are cellularly covered by the restrictions of the \(f_i\) to the necessarily finite 2-skeleton of \(K\).
Wise answers the above question in the negative. He showed [D. T. Wise, Comment. Math. Helv. 82, No. 4, 683–724 (2007; Zbl 1142.20025)] that there is a pair \((K_1,K_2)\) of finite 2-dimensional CW complexes which have a common infinite cellular cover \(K\), but which have no finite CW complex which cellularly covers both. Call such a pair a non-Leighton pair.
Wise’s non-Leighton pairs contain as few as six 2-cells. The challenge then becomes that of determining how “small” 2-dimensional cellular non-Leighton pairs can be. That is, what is the minimum number of cells, or more specifically, the minimum number of 2-cells, possible in a 2-dimensional non-Leighton CW complex pair.
The authors of the manuscript under review prove
Theorem. The standard 2-dimensional CW complexes associated to the presentations \[ H_{\epsilon} = \langle a,c,d: a^{[c^d,c]}=a^{\epsilon},(c^3)^d=c^5 \rangle , \] \(\epsilon = \pm 1\), containing one vertex, three 1-cells, and two 2-cells, are a non-Leighton pair. That is, they have a common cellular covering, but no common finite cellular covering.
The \(H_{\epsilon}\) are amalgamated products of the fundamental group of the torus \(BS(1,1)\) and the Baumslag-Solitar group \(BS(3,5)\), amalgamated over a cyclic group.
The question of whether there exist non-Leighton pairs of 2-dimensional CW complexes with only one 2-cell is, at this time, still open.
Reviewer: Mathew Timm (Peoria)
MSC:
| 57K20 | 2-dimensional topology (including mapping class groups of surfaces, Teichmüller theory, curve complexes, etc.) |
| 20F05 | Generators, relations, and presentations of groups |
| 05C65 | Hypergraphs |
| 05C25 | Graphs and abstract algebra (groups, rings, fields, etc.) |
References:
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