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On four-dimensional s-cobordisms. II. (English) Zbl 0685.57011

[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

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

Citations:

Zbl 0597.57008
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