Cappell, Sylvain E.; Shaneson, Julius L. On four-dimensional s-cobordisms. II. (English) Zbl 0685.57011 Comment. Math. Helv. 64, No. 2, 338-347 (1989). [For Part I see J. Differ Geom. 22, 97-115 (1985; Zbl 0597.57008).] Let \(Q_ r\) be the quaternionic group of order \(2^{r+2}\) and \(M_ r=S^ 3/Q_ r\) be the quotient of the canonical action. The main theorem of the paper says that there exist precisely \(2^{2^ r-r-1}\) distinct topological s-cobordisms of M, to itself. The claim is reduced to the case \(r=1\). Using Freedman’s topological surgery sequence the proof is reduced to the verification that the surgery obstruction \(\xi \in L_ 5(Q_ r)\) of a specific surgery problem over \(M_ r\times I\times I\) lies not in the image of the map \(\theta\) : [\(\Sigma^ 2M_+;G/TOP]\to L_ 5(Q_ r)\). This is done using the visible symmetric L-groups introduced by M. Weiss. Reviewer: W.Lück Cited in 4 Documents MSC: 57N70 Cobordism and concordance in topological manifolds 57R67 Surgery obstructions, Wall groups 57N13 Topology of the Euclidean \(4\)-space, \(4\)-manifolds (MSC2010) 57S25 Groups acting on specific manifolds 57S17 Finite transformation groups 57N10 Topology of general \(3\)-manifolds (MSC2010) 57R80 \(h\)- and \(s\)-cobordism Keywords:topological 4-dimensional s-cobordism; 3-dimensional; quaternionic space form; quaternionic group of order \(2^{r+2}\); topological s-cobordisms; Freedman’s topological surgery sequence; surgery obstruction; visible symmetric L-groups Citations:Zbl 0597.57008 PDFBibTeX XMLCite \textit{S. E. Cappell} and \textit{J. L. Shaneson}, Comment. Math. Helv. 64, No. 2, 338--347 (1989; Zbl 0685.57011) Full Text: DOI EuDML