Vanishing of Whitehead torsion in dimension four
Let \(M\) be a compact oriented 3-manifold. It is called geometric if it can be cut by embedded spheres and incompressible tori into pieces which are Seifert or hyperbolic. Thurston's geometrization conjecture says that \(M\) is always geometric and known to be true if \(M\) is Haken. An \(h\)- cobordism \((W;M_ 0,M_ 1)\) is a compact manifold whose boundary is the disjoint union of \(M_ 0\) and \(M_ 1\) such that the inclusions of both \(M_ 0\) and \(M_ 1\) into \(W\) are homotopy equivalences. It is called an \(s\)-cobordism if the Whitehead torsion of the inclusion \(M_ 0\) into \(W\) is trivial. The Whitehead torsion takes value in the Whitehead group \(\text{Wh}(\pi_ 1(M_ 0))\). Any element in the Whitehead group can be realized as a Whitehead torsion of an \(h\)-cobordism over \(M_ 0\) and any \(s\)-cobordism is homeomorphic to the cylinder \(M_ 0\times [0,1]\)) provided \(\dim(W)\geq 6\). The paper deals with the 4-dimensional \(h\)-cobordisms. The main theorem says for an \(h\)-cobordism \((W,M_ 0,M_ 1)\) between geometric closed orientable 3-manifolds \(M_ 0\) and \(M_ 1\) that it is an \(s\)-cobordism. In particular the theorem applies if \(M_ 0\) and \(M_ 1\) are 3- dimensional linear space forms. The main techniques used in the proof are topological surgery as developed by Freedman and proper surgery theory.
scientific article; zbMATH DE number 167780