Groupoid extensions of mapping class representations for bordered surfaces
Let us first quote the authors' abstract: ``The mapping class group of a surface with one boundary component admits numerous interesting representations including a representation as a group of automorphisms of a free group and as a group of symplectic transformations. Insofar as the mapping class group can be identified with the fundamental group of Riemann's moduli space, it is furthermore identified with a subgroup of the fundamental path groupoid upon choosing a basepoint. A combinatorial model for this, the mapping class groupoid, arises from the invariant cell decomposition of Teichmüller space, whose fundamental path groupoid is called the Ptolemy groupoid. It is natural to try to extend representations of the mapping class group to the mapping class groupoid, i.e., to construct a homomorphism from the mapping class groupoid to the same target that extends the given representations arising from various choices of basepoint. Among others, we extend both aforementioned representations to the groupoid level in this sense, where the symplectic representation is lifted both rationally and integrally. The techniques of proof include several algorithms involving fatgraphs and chord diagrams. The former extension is given by explicit formulae depending upon six essential cases, and the kernel and image of the groupoid representation are computed. Furthermore, this provides groupoid extensions of any representation of the mapping class group that factors through its action on the fundamental group of the surface including, for instance, the Magnus representation and representation on the moduli spaces of flat connections.'' Let us describe the content of the paper in more detail. Section~1 is an introduction. In Section~2 the notion of fatgraph is reminded. Fatgraph is a graph with a cyclic ordering of oriented edges pointing to every fixed vertex. Boundary cycles of a fatgraph and its genus are defined naturally. A (once-)bordered fatgraph is a fatgraph with a unique boundary cycle such that all its vertices are at least trivalent except for a unique univalent one. For every bordered fatgraph there is a natural linear ordering on the set of all oriented edges of \(G\) induced by transversing its boundary cycle. Let \(\Sigma_{g,1}\) be an orientable surface of genus \(g\) with one boundary component. A Whitehead move is a transformation of a fatgraph which contracts some edge into a vertex and then replaces the vertex by an edge in such a way that the new fatgraph is not isomorphic to the initial one. The mapping class groupoid \(\mathfrak{MC}(\Sigma_{g,1})\) is defined as the set of finite compositions of Whitehead moves on bordered fatgraphs modulo pentagon, commutativity, and involving relations. A marking of a bordered fatgraph \(G\) is an isotopy class of embeddings of \(G\) into \(\Sigma_{g,1}\) with some natural conditions on these embeddings. The Ptolemy groupoid \(\mathfrak{Pt}(\Sigma_{g,1})\) is the set of finite compositions of Whitehead moves on marked bordered fatgraphs of genus \(g\) modulo pentagon, commutativy, and involving relations. In Section~3 an algorithm is described to determine a canonical maximal tree \(T_G\) in each bordered fatgraph \(G\). The complement of \(T_G\) is the linearly ordered set of oriented edges, \(X_G\), it is named the set of generators. For a marked bordered graph, \(X_G\) defines the natural ordered set of generators of \(\pi_1:=\pi_1(\Sigma_{g,1})\) (Theorem 3.3). Then for each Whitehead move \(W\) a canonically associated element \(\widetilde{N}(W)\in Aut(\pi_1)\) is defined. It is natural in the sense that if \(\varphi\) is represented via a sequence of Whitehead moves \(W_i\) then \(\widetilde{N}(W_i)\) agrees with the image \(N(\varphi)\) of \(\varphi\) under the Nielsen embedding \(N\) of the mapping class group \(MC(\Sigma_{g,1})\) into \(Aut(\pi_1)\) (Corollary~3.5). There are six essential cases of Whitehead moves. It is investigated how they change the set \(X_G\). As a result, we have Theorem~3.6 which states that there is an explicit extension \(\widehat{N}: \mathfrak{MC}(\Sigma_{g,1})\to Aut(F_{2g})\) of Nielsen's embedding to a representation of the mapping class groupoid with target \(Aut(F_{2g})\). In Section~4 the Magnus representation of \(Aut(\pi_1)\) is considered. It is associated with the Fox Jacobian of the element from \(Aut(\pi_1)\) with respect to a given basis. Corollary 3.5 implies that the Magnus representation extends to the Ptolemy groupoid (with target \(Gl(2g,\mathbb{Z}[\pi_1]))\). In Section 5, making use of the technique of chord diagrams, it is proved that the kernel of the extension \(\widetilde{N}:\mathfrak{Pt}(\Sigma_{g,1})\to Aut(\pi_1)\) is generated by Whitehead moves of the first two types. In Section 6 the image of \(\widetilde{N}\) is described. Moreover, it is proved that there is an explicit extension \(\widetilde{id}:\mathfrak{Pt}(\Sigma_{g,1})\to MC(\Sigma_{g,1})\) of the identity homomorphism of \(MC(\Sigma_{g,1})\) (Theorem 6.1). For the symplectic representation of \(MC(\Sigma_{g,1})\) a similar statement it is proved as well (Corollary 6.2). In Section 7, Theorem 7.2 is proved. It states that there is an explicit extension \(\widehat{\tau}_0: MC(\Sigma_{g,1}) \to Sp(2g,\mathbb{Z})\) to the mapping class groupoid of the symplectic representation of the mapping class group. At last in Section 8 it is proved that there are explicit extensions of identity representations of \(\mathfrak{Pt}(\Sigma_{g,1})\) in the Torelli group and in the subgroup \(MC(\Sigma_{g,1})\) of mapping classes preserving the Lagrangian.
scientific article