Branched coverings of 2-complexes and diagrammatic reducibility

Abstract

Abstract.The condition that all spherical diagrams in a 2-complex be reducible is shown to be equivalent to the condition that all finite branched covers be aspherical. This result is related to the study of equations over groups. Furthermore large classes of 2-complexes are shown to be diagrammatically reducible in the above sense; in particular, every 2-complex has a subdivision which admits a finite branched cover which is diagrammatically reducible.A consequence of the classical Riemann-Hurwitz formula for Riemann surfaces [6, p. 301] is the fact that a finite branched cover of an aspherical Riemann surface is also aspherical. However the property of asphericity of a 2-complex implying asphericity of finite branched covers fails for general 2-complexes. In fact we shall prove (Theorem 4.5 below) that this property holds precisely for those 2-complexes which are diagrammatically reducible.A 2-complex A is said to be diagrammatically reducible (abbreviated DR( X)) if for every combinatorial map /: S2 -* X of the 2-sphere into X (a "spherical diagram" in X) there is a pair of faces in the domain with an edge in common which are mapped mirrorwise by / across that edge. A closely related notion was introduced by Lyndon and Schupp [13] but this notion appears in its present form in a paper by Sieradski [14]. In that same paper Sieradski asked a question which would have implied that all classical knot complements in the 3-sphere are homotopy equivalent to diagrammatically reducible 2-complexes. We shall prove this latter assertion (Theorem 6.5) making use of our amalgamation theorem for diagrammatic reducibility (Theorem 5.4).The characterization of diagrammatic reducibility in terms of branched covers follows from a lifting theorem which may be of general interest. If /: X -> Y is a combinatorial map of combinatorial 2-complexes, then / induces a map of the link complexes of vertices Lf: Lx -* LY (the "star graphs" or "coinitial graph" in group theoretical language). The map / is said to be reduced if L, is an immersion. We prove (Theorem 4.2) that if /: X -» Y is a reduced map of 2-complexes with A finite, then there is a commutative diagram (see below) where ir. Z -* Y is a finite branched cover and where g is injective off the zero skeleton of X. This result is

Isometric embeddings of surfaces for scl

“…The number of vertices in this subdivision of S 1 σ is called the degree of σ. This follows Gersten's terminology [15].…”

“…Each vertex v in a 2-complex X has a link -denoted by Lk X (v) -which can be defined as the boundary of a regular neighbourhood of v in X (see [15]). The link has the structure of a graph, with vertices of Lk X (v) corresponding to oriented halfedges e of X starting at v, with an edge between e 1 and e 2 in Lk X (v) corresponding to each 2-cell σ of X whose boundary traverses e −1…”