Abstract:
Let W be the exterior of a knot in a homology sphere and let M be an amalgamation of W and any other compact 3-manifold along boundary torus. Let N be the manifold obtained by pinching W into a solid torus. This means that there is a degree-one map from M to N. We prove that the Heegaard genus of M is at least as large as the Heegaard genus of N. An immediate corollary is that the tunnel number of a satellite knot is at least as large as the tunnel number of its pattern knot.