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In a remarkable series of papers, both anticipating development in geometric topology and algebraic K-theory, specifically what we call now the Farrell-Jones conjecture, Waldhausen introduced obstructions to the Whitehead groups to satisfy a mayer-vietoris sequence. See https://mathscinet.ams.org/mathscinet-getitem?mr=498807

These obstructions are related to splitting of homotopy equivalences across submanifolds of high dimension ( >5) due to Sylvain Cappell: https://mathscinet.ams.org/mathscinet-getitem?mr=285010.

Since then , lots of effort has been put into showing vanishing results for Waldhausen Nil groups. I would like to know an easy example of non-vanishing of these groups. It would be interesting to know the minimal cohomological dimension of the groups involved in an amalgam of such example. Waldhausen sorts out in his original paper fundamental groups of surfaces, free groups, fundamental groups of submanifolds of the three dimensional spheres...

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There are examples of non-vanishing nil groups due to Daniel Juan-Pineda. See Juan-Pineda, Daniel(MEX-NAMMO-IM) On higher nil groups of group rings. Homology Homotopy Appl. 9 (2007), no. 2, 95–100 (MSN).

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