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Graphic descriptions of monodromy representations. (English) Zbl 1123.57004

A graphical method is proposed in order to describe an arbitrary monodromy representation \(\rho: \pi_1(\Sigma - \Delta) \to G\), where \(\Sigma\) is an oriented surface with (possibly empty) connected boundary, \(\Delta \subset \mathop{\text{Int}} \Sigma\) is a finite set of punctures on \(\Sigma\) and \(G\) is any group. The description is provided in terms of a suitably decorated oriented graph \(\Gamma \subset \Sigma\).
More precisely, for a presentation \(G \cong \langle \mathcal X, \mathcal R \rangle\) and a set \(\mathcal S\) of words on \(\mathcal X \cup \mathcal X^{-1}\), \(\Gamma\) is called an \((\mathcal X, \mathcal R, \mathcal S)\)-chart, if its vertices are colored black or white and its edges are labelled by elements of \(\mathcal X\), in such a way that each small positive loop around a black (resp. white) vertex in \(\mathop{\text{Int}} \Sigma\) gives rise, up to cyclic permutation, to an intersection word in \(\mathcal S\) (resp. \(\mathcal R \cup \mathcal R^{-1}\)). Such a chart describes a monodromy \(\rho\) as above, with \(\Delta\) consisting of black vertices of \(\Gamma\) and \(\rho(\lambda)\) defined by the intersection word of \(\lambda\). In particular, the monodromy of any positive meridian around a puncture is represented by a word in \(\mathcal S\).
A chart calculus is also introduced, in terms of certain moves relating any two charts describing the same monodromy or conjugate monodromies with the same set of punctures \(\Delta\), as well as monodromies which are equivalent up to isotopy or homeomorphism of \(\Sigma\).
Special cases are discussed, where \(G = B_m\) (the braid group of order \(m\)) or \(G = \mathop{\text{SL}}(2,\mathbb Z)\) (the mapping class group of the torus), respectively corresponding to the monodromy representations of braided surfaces with \(m\) sheets in \(\Sigma \times D^2\) and to genus \(1\) Lefschetz fibrations over \(\Sigma\).

MSC:

57M20 Two-dimensional complexes (manifolds) (MSC2010)
57M12 Low-dimensional topology of special (e.g., branched) coverings
57M15 Relations of low-dimensional topology with graph theory
57Q45 Knots and links in high dimensions (PL-topology) (MSC2010)
58K10 Monodromy on manifolds
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References:

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