Let \(M\) be a closed orientable 3-manifold. A genus \(g\) Heegaard splitting of \( M\) is a decomposition of \(M\) into two genus \(g\) handlebodies \(H_1\) and \(H_2\) with a common boundary, that is, \(M=H_1\cup H_2\), and \(H_1\cap H_2=\partial H_1 =\partial H_2\). A Heegaard splitting is described by an ordered triple \((H_1,H_2,S)\), where \(S=\partial H_1 =\partial H_2\) is called a Heegaard surface. Two Heegaard splittings \((H_1,H_2,S)\) and \((H_1',H_2',S')\) of \(M\) are called equivalent if there is an ambient isotopy of \(M\) that carries \((H_1,H_2,S)\) to \((H_1',H_2',S')\). A stabilization of a genus \(g\) Heegaard surface \(S\) of \(M\) is a surface of genus \(g+1\) obtained by adding a trivial 1-handle to \(S\), that is, a 1-handle whose core is parallel to \(S\). The new surface splits \(M\) into two genus \(g+1\) handlebodies, then giving a new Heegaard decomposition of \(M\).
It is known, by old results of K. Reidemeister and J. Singer, that any two Heegaard splittings of a 3-manifold \(M\) become equivalent after a finite sequence of stabilizations. It was a long standing conjecture, called the stabilization conjecture (see Problem 3.89 in R. Kirby’s list [Problems in low-dimensional topology, AMS/IP Stud. Adv. Math. 2 (pt. 2), 35–473 (1997; Zbl 0888.57014)]), that a single stabilization suffices to make two splittings equivalent, that is, if \(S_p\) and \(S_q\) are two Heegaard surfaces of \(M\) of genus \(p\) and \(q\), \(p\leq q\), then by making a single stabilization in \(S_q\) and \(q-p+1\) stabilizations in \(S_p\) one gets equivalent splittings. All previously known examples verify the conjecture.
In the main result of the paper under review it is shown that the conjecture is false, namely, it is shown that for each \(g\geq 2\) there is a 3-manifold \(M_g\) with two genus \(g\) Heegaard splittings that require \(g\) stabilizations to become equivalent. Roughly speaking, the proof is as follows. Begin with a hyperbolic manifold \(M_\phi\) that fibers over the circle with fiber a genus \(g\) surface \(S_\phi\). Cut open \(M_\phi\) along \(S_\phi\) to get a manifold \(B\), and let \(L\) and \(R\) be two manifolds obtained by taking \(n\) copies of \(B\) glued end to end. So \(L\), \(R\) are homeomorphic but not isometric to a product. Let \(M\) be a 3-manifold with certain metric properties, separated into two genus \(g\) handlebodies \(H_L\) and \(H_R\). Insert \(L\cup R\) in the middle of \(M\), getting a new manifold \(M_g\), which has two Heegaard splittings, \(E_0=(H_L\cup L, H_R\cup R,S)\) and \(E_1=(H_R\cup R,H_L\cup L,-S)\), where \(S\) is the common boundary of \(H_L\cup L\) and \(H_R\cup R\) and \(-S\) is \(S\) with the opposite orientation. Let \(G_0\) and \(G_1\) be the Heegaard splittings obtained by making \(g-1\) stabilizations to \(E_0\) and \(E_1\). Each splitting \(G_0\), \(G_1\) defines a family of surfaces that sweep out \(M_g\) from the spine of one handlebody to the spine of the other handlebody.
It is shown that this family of surfaces can be deformed to a family of harmonic maps, which have area that is uniformly bounded by a constant that is independent of \(n\). Then it is shown that for \(n\) sufficiently large, the surfaces in such bounded area families cannot interpolate between \(G_0\) and \(G_1\), so \(E_0\) and \(E_1\) require at least \(g\) stabilizations to become equivalent; in fact, these splittings are equivalent after \(g\) stabilizations. A similar result has been announced independently and with different techniques by D. Bachman.