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Branched coverings of 2-complexes and diagrammatic reducibility. (English) Zbl 0644.20023
In 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.}
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
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