A non-Abelian \(K\)-theory and pseudo-isotopies of 3-manifolds (Q1915493)

In [\textit{A. E. Hatcher} and \textit{J. B. Wagoner}, Astérisque 6, 8-238 (1973; Zbl 0274.57010)] it is shown how the components of the space of pseudo-isotopies of a compact manifold of dimension at least seven can be determined by invariants of an algebraic \(K\)-theoretic nature with the main invariant in an abelian group \(Wh_2(\pi_1 M)\) which arises from a study of automorphisms of free f.g. \([\pi_1 M]\)-modules. The present paper begins a similar approach to a study of pseudo-isotopies of 3-dimensional manifolds. On the algebraic side, this involves non-commutative versions \(KN_2(\pi)\) and \(WhN_2(\pi)\) of \(K_2(\pi)\) and \(Wh_2(\pi)\). The definition of these follow the approach in [\textit{J. Milnor}, Introduction to algebraic \(K\)-theory, Ann. Math. Stud., No. 72 (1971; Zbl 0237.18005)], but with \([\pi]^n\) replaced by a certain non-Abelian group \(F_n[\pi]\) where \(F_n\) is the free group on \(\{x_1, x_2,\dots, x_n\}\). Special cases of this part of the paper have some overlap with [\textit{S. M. Gersten}, J. Pure Appl. Algebra 33, 269-279 (1984; Zbl 0542.20021)]. When \(n\) is fixed, one also gets unstable groups \(KN_2(n,\pi)\) and \(WhN_2(n,\pi)\) as well as a certain extension \(KN_2'(n,\pi) \twoheadrightarrow KN_2(n,\pi)\) (which may be an isomorphism). On the geometric side, the main result is Theorem 4.2 which expresses \(KN_2'(g,\pi_1(M))\) in terms of a certain space of Morse functions \(f:(M\times I;M\times 0,M\times 1)\to(I;0,1)\) with precisely \(g\) critical points of index one. Here \(M\) is a 3-manifold with no fake 3-balls. Theorems of this sort can be, and frequently are, presented as a reduction of a geometric problem to an algebraic one. However, one often sees the actual information flow going from geometry to algebra. This is also the case in the present paper, where the above mentioned Theorem 4.2 is combined with geometric arguments to get the main (algebraic) result, namely the vanishing of \(WhN_2(n,\pi)\) whenever \(\pi\) is an irreducible 3-manifold group. The paper is quite technical, and the reviewer finds that the perusal of it could (and should) have been made easier by a more careful exposition. In particular, the statement of Theorem 4.2 involves a based pair of spaces \(({\mathcal S}_g,{\mathcal T}_g;h_g)\) the definition of which ought to have been highlighted, e.g. by labelling it.