\(K_ 2\) invariants of 3-dimensional pseudoisotopies (Q1902015)

For an \(n\)-dimensional manifold \(M\) let \({\mathcal C}(M)\) denote its topological concordance space. By work of Hatcher, Wagoner and Igusa, \(\pi_0({\mathcal C}(M))\) is understood in terms of higher algebraic \(K\)-theory for \(n\geq 5\). In particular, for \(n\geq 5\), the map \(\sigma: \pi_0(M)\to Wh_2\mathbb{Z}[\pi]\) is surjective where \(\pi=\pi_1(M)\). The authors analyze here the image \(\text{Im} (\sigma)\) for \(n=3\). When \(\pi\) is infinite then \(Wh_2\mathbb{Z}[\pi]\) is conjectured to be zero. The authors therefore concentrate on spaces with finite fundamental group and compute \(\text{Im}(\sigma)\) in the case of 3-dimensional lens space (with \(\pi\) of odd order) and linear quaternionic space form. In particular, they show that \(\sigma\) is not surjective for certain 3-manifolds. Hence, unlike in higher dimensions, not all elements in \(Wh_2\mathbb{Z}[\pi]\) are realized by pseudoisotopies. As part of their treatment, the authors give also a proof of the old conjecture/folk theorem that \(K_2\mathbb{Z}[\pi]\) is finite when \(\pi\) is finite.