Amalgamations of Heegaard splittings in 3-manifolds without some essential surfaces
Let \(M\) be a compact, orientable, boundary irreducible, 3-manifold and \(F\) be a connected essential surface in \(M\) with genus \(g(F) \geq 1\) which cuts \(M\) into \(M_1\) and \(M_2\). The authors prove that \(g(M) = g(M_1) + g(M_2) - g(F)\) provided there do not exist any essential surfaces with boundary \((Q_i, \partial Q_i)\) in \((M_i,F)\) that satisfy \(\chi(Q_i) > 2 + g(F) - 2g(M_i)\) for \(i=1,2\). As a consequence they prove that if \(M_i\) has a Heegaard splitting \(V_i \cup_{S_i} W_i\) with distance satisfying \[ D(S_i) \geq 2 g(M_i) - g(F) \] for \(i=1,2\), then \[ g(M) = g(M_1) + g(M_2) - g(F). \] The distance of a Heegaard splitting was defined by \textit{J. Hempel} in [Topology 40, No.~3, 631--657 (2001; Zbl 0985.57014)].
scientific article; zbMATH DE number 5624198
scientific article