Disk complexes and genus two Heegaard splittings for non-prime 3-manifolds

Given a genus two Heegaard splitting for a non-prime 3-manifold, we define a special subcomplex of the disk complex for one of the handlebodies of the splitting, and then show that it is contractible. As applications, first we show that the complex of Haken spheres for the splitting is contractible, which refines the results of Lei and Lei-Zhang. Secondly, we classify all the genus two Heegaard splittings for non-prime 3-manifolds, which is a generalization of the result of Montesinos-Safont. Finally, we show that the mapping class group of the splitting, called the Goeritz group, is finitely presented by giving its explicit presentation.