Contractibility of the Kakimizu complex and symmetric Seifert surfaces (Q2880688)

The Kakimizu complex \(MS(L)\) of a link \(L\) was first defined by \textit{O. Kakimizu} [Hiroshima Math. J. 22, 225--236 (1992; Zbl 0774.57006)]. This simplicial complex has a vertex for each ambient isotopy class of minimal genus Seifert surfaces for \(L\). Distinct vertices are adjacent if they can be realised disjointly. The complex is a flag complex (that is, any finite set of mutually adjacent vertices spans a simplex).NEWLINENEWLINEThe authors point out that an alternative criterion for determining the adjacency of pairs of vertices is more suitable. This criterion considers lifts of the Seifert surfaces to the infinite cyclic cover of the link complement. The resulting complex is only changed in the case of links with disconnected Seifert surfaces. The authors then generalize this definition, defining a complex \(MS(E,\gamma,\alpha)\) for suitable choices of a 3-manifold \(E\), a set \(\gamma\) of simple closed curves on \(\partial E\), and a homology class \(\alpha\) with \(\partial \alpha = [\gamma]\).NEWLINENEWLINEFirst it is shown that the Kakimizu complex is contractible. The main tool for doing so is a partial ordering of the vertices that is defined using the infinite cyclic cover. The authors then address the more general subject of fixed point sets under subgroups of the mapping class group. The main result here is as follows. Let \(G\) be any subgroup of the mapping class group of \(E\) fixing \(\alpha\) and the homotopy class of \(\gamma\). Then its fixed-point set is either empty or contractible.