On 4-dimensional s-cobordisms
The authors exhibit a family of orientable counterexamples to the topological s-cobordism theorem in dimension 4. Let \(Q_ r\) be the quaternion group of order \(2^{r+1}:\) \(Q_ r=\{y,x|\) \(y^ 2=x^{2r}\), \(yxy^{-1}=x^{-1}\}\) with \(r\geq 2\). Let \(M_ r\) be a 3- dimensional spherical space-form with fundamental group \(Q_ r\). For each invertible knot \(k\subset S^ 3\) whose \(2^{r+1}\)-fold branched cyclic cover is a homology sphere the authors construct a compact topological 4-manifold \(W_ k\) with boundary \(M_ r\times \{0,1\}\) so that \(W_ k\) satisfies the hypotheses of the s-cobordism theorem. The construction crucially uses Freedman's realization of a hyperbolic plane in \(\pi_ 2(X)\) where X is a 4-manifold with finite \(\pi_ 1\) by an imbedded punctured \(S^ 2\times S^ 2\). It is shown that if \(W_ k\) is homeomorphic to \(M_ r\times [0,1]\) then \(\Delta_ k(t)\equiv u^ 2\) mod (t\({}^ s-1)\) where \(\Delta_ k\) is the Alexander polynomial of k, u is an element of Z[t] and \(s=2^{r+1}\). For example, if k is the torus knot of type \((2^ r-1,2^ r+1)\) then the condition is not satisfied and hence \(W_ k\) is not \(M_ r\times [0,1]\). The proof of the condition on \(\Delta_ k\) uses the technique of Reidemeister torsions and the fact (due independently to the authors and H. Rubinstein) that homotopic self-homeomorphisms of \(M_ r\) are actually isotopic.
counterexamples to the topological s-cobordism theorem in
counterexamples to the topological s-cobordism theorem in dimension 4
scientific article; zbMATH DE number 3962768