Branched coverings of 2-complexes and diagrammatic reducibility.

*(English)*Zbl 0644.20023In a seminal paper [Math. Ann. 166, 208-228 (1966; Zbl 0138.257)] R. Lyndon introduced a notion of a reduced diagram over a presentation of a group, and, as is well known, this and similar concepts may as well be phrased over a combinatorial 2-complex. In the language of D. Collins and the reviewer [Math. Ann. 261, 155-183 (1982; Zbl 0477.20019)], referred to henceforth as [CH], a diagram (M,\(\psi)\) over a combinatorial 2-complex X consists of a singular multidisk \((M_ 0,\psi_ 0)\) and several singular spheres (possibly tailed) \((S_ i,\psi_ i)\) over X; here \(M=\cup M_ i\), each \(\psi_ i\), \(i\geq 0\), is a morphism of combinatorial 2-complexes, and \(\psi\) introduces “labels” for the vertices, edges and faces of M; see [CH] for details. To explain Lyndon’s notion of a reduced diagram, let (M,\(\psi)\) be a diagram over X, and consider the following:

Property 1. M has two distinct faces \(\Delta_ 1\) and \(\Delta_ 2\) and an edge e such that, for suitable \(\alpha\) and \(\beta\), \(\partial \Delta_ 1=\alpha e\), \(\partial \Delta_ 2=e^{-1}\beta\), and the product of the labels on \(\alpha\) and \(\beta\) is freely trivial.

A diagram (M,\(\psi)\) is reduced in the sense of Lyndon [loc. cit.] if it does not have property 1; we note that in Lyndon’s paper the labels assume their values in the corresponding free group. In the paper under review a combinatorial 2-complex X is said to be diagrammatically reducible if every diagram (S,\(\psi)\) over X where S is a tesselated 2- sphere has the following:

Property 2. S has two distinct faces F and F’ having an edge e in common, and there is an orientation reversing homeomorphism \(g: F\to F'\) fixing \(F\cap F'\) pointwise so that \(\psi | F'=\psi | F\circ g.\)

Among others it is proved (Theorem 4.5) that for a combinatorial 2- complex X the following assertions are equivalent: (1) X is diagrammatically reducible; (2) every finite branched covering of X is aspherical; (3) every finite branched covering of X is Cockcroft.

Here a branched covering is a combinatorial map \(f: Y\to X\) whose restriction to \(Y-Y^ 0\) is a finite sheeted covering, where \(Y^ 0\) denotes the zero skeleton. This result is then related to equations over groups. Furthermore, by means of an amalgamation technique, examples of diagrammatically reducible 2-complexes are given, e.g. 2-complexes arising from bounded Haken manifolds; this answers a question of A. Sieradski [Q. J. Math., Oxf. II. Ser. 34, 97-106 (1983; Zbl 0522.57003)].

{Reviewer’s remarks: Diamond moves were introduced in [CH] and not in [I. M. Chiswell, D. J. Collins, J. Huebschmann, Math. Z. 178, 1-36 (1981; Zbl 0443.20030)], as asserted on p. 705 of the paper. Moreover, the notion of diagrammatic reducibility of combinatorial 2- complexes in the paper under review uses non-combinatorial concepts such as e.g. “homeomorphism” and “fixing pointwise”. Here is an entirely combinatorial description: As in 1.1 of [CH], define the boundary path \(\partial \Delta\) of a face \(\Delta\) to be some designated closed path so that \(\partial (\Delta^{-1})=(\partial \Delta)^{-1}\), and define the initial vertex of \(\partial \Delta\) to be the basepoint of \(\Delta\), written v(\(\Delta)\). Given a diagram (M,\(\psi)\) over X, consider the following:

Property 3. (M,\(\psi)\) satisfies property 1 in such a way that \(v(\Delta_ 1)=v(\Delta_ 2).\)

Further, call (M,\(\psi)\) based if \(\psi\) preserves basepoints. One can then show that a combinatorial 2-complex X is diagrammatically reducible if and only if every based diagram (S,\(\psi)\) over X where S is a tesselated 2-sphere has property 3.

Since the term reducible diagram used in the paper under review was also used in [CH], we indicate the differences between the two. In the paper under review, reducibility for a diagram is a stricter condition than the notion in [CH] because in [CH] certain “reducing” operations on diagrams are permitted which are not allowed here. Also in [CH], a 2- complex was called diagrammatically aspherical if every spherical diagram over X is reducible. Consequently a diagrammatically reducible 2-complex X, as defined by Gersten, is diagrammatically aspherical. A diagrammatically reducible 2-complex can also be shown to be aspherical whereas a diagrammatically aspherical complex need not be aspherical. Whether an aspherical, diagrammatically aspherical 2-complex must actually be diagrammatically reducible is unknown.}

Property 1. M has two distinct faces \(\Delta_ 1\) and \(\Delta_ 2\) and an edge e such that, for suitable \(\alpha\) and \(\beta\), \(\partial \Delta_ 1=\alpha e\), \(\partial \Delta_ 2=e^{-1}\beta\), and the product of the labels on \(\alpha\) and \(\beta\) is freely trivial.

A diagram (M,\(\psi)\) is reduced in the sense of Lyndon [loc. cit.] if it does not have property 1; we note that in Lyndon’s paper the labels assume their values in the corresponding free group. In the paper under review a combinatorial 2-complex X is said to be diagrammatically reducible if every diagram (S,\(\psi)\) over X where S is a tesselated 2- sphere has the following:

Property 2. S has two distinct faces F and F’ having an edge e in common, and there is an orientation reversing homeomorphism \(g: F\to F'\) fixing \(F\cap F'\) pointwise so that \(\psi | F'=\psi | F\circ g.\)

Among others it is proved (Theorem 4.5) that for a combinatorial 2- complex X the following assertions are equivalent: (1) X is diagrammatically reducible; (2) every finite branched covering of X is aspherical; (3) every finite branched covering of X is Cockcroft.

Here a branched covering is a combinatorial map \(f: Y\to X\) whose restriction to \(Y-Y^ 0\) is a finite sheeted covering, where \(Y^ 0\) denotes the zero skeleton. This result is then related to equations over groups. Furthermore, by means of an amalgamation technique, examples of diagrammatically reducible 2-complexes are given, e.g. 2-complexes arising from bounded Haken manifolds; this answers a question of A. Sieradski [Q. J. Math., Oxf. II. Ser. 34, 97-106 (1983; Zbl 0522.57003)].

{Reviewer’s remarks: Diamond moves were introduced in [CH] and not in [I. M. Chiswell, D. J. Collins, J. Huebschmann, Math. Z. 178, 1-36 (1981; Zbl 0443.20030)], as asserted on p. 705 of the paper. Moreover, the notion of diagrammatic reducibility of combinatorial 2- complexes in the paper under review uses non-combinatorial concepts such as e.g. “homeomorphism” and “fixing pointwise”. Here is an entirely combinatorial description: As in 1.1 of [CH], define the boundary path \(\partial \Delta\) of a face \(\Delta\) to be some designated closed path so that \(\partial (\Delta^{-1})=(\partial \Delta)^{-1}\), and define the initial vertex of \(\partial \Delta\) to be the basepoint of \(\Delta\), written v(\(\Delta)\). Given a diagram (M,\(\psi)\) over X, consider the following:

Property 3. (M,\(\psi)\) satisfies property 1 in such a way that \(v(\Delta_ 1)=v(\Delta_ 2).\)

Further, call (M,\(\psi)\) based if \(\psi\) preserves basepoints. One can then show that a combinatorial 2-complex X is diagrammatically reducible if and only if every based diagram (S,\(\psi)\) over X where S is a tesselated 2-sphere has property 3.

Since the term reducible diagram used in the paper under review was also used in [CH], we indicate the differences between the two. In the paper under review, reducibility for a diagram is a stricter condition than the notion in [CH] because in [CH] certain “reducing” operations on diagrams are permitted which are not allowed here. Also in [CH], a 2- complex was called diagrammatically aspherical if every spherical diagram over X is reducible. Consequently a diagrammatically reducible 2-complex X, as defined by Gersten, is diagrammatically aspherical. A diagrammatically reducible 2-complex can also be shown to be aspherical whereas a diagrammatically aspherical complex need not be aspherical. Whether an aspherical, diagrammatically aspherical 2-complex must actually be diagrammatically reducible is unknown.}

Reviewer: J.Huebschmann

##### MSC:

20F05 | Generators, relations, and presentations of groups |

57M20 | Two-dimensional complexes (manifolds) (MSC2010) |

57M12 | Low-dimensional topology of special (e.g., branched) coverings |