Heegaard genus, degree‐one maps, and amalgamation of 3‐manifolds

Abstract

Let M = W ∪ T V $M=\mathcal {W}\cup _\mathcal {T} \mathcal {V}$ be an amalgamation of two compact 3-manifolds along a torus, where W $\mathcal {W}$ is the exterior of a knot in a homology sphere. Let N $N$ be the manifold obtained by replacing W $\mathcal {W}$ with a solid torus such that the boundary of a Seifert surface in W $\mathcal {W}$ is a meridian of the solid torus. This means that there is a degree-one map f : M → N $f\colon M\rightarrow N$ , pinching W $\mathcal {W}$ into a solid torus while fixing V $\mathcal {V}$ . We prove that g ( M ) ⩾ g ( N ) $g(M)\geqslant g(N)$ , where g ( M ) $g(M)$ denotes the Heegaard genus. 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.