Pseudoisotopies and the Bökstedt trace
The author studies the space \(\mathcal P(M)\) of smooth pseudo-isotopies of the compact smooth manifold \(M\). A motivation of the paper is the question whether \(\mathcal P(M)\) is disconnected whenever \(M\) is non-simply-connected. The author shows that \(\mathcal P(M)\) is disconnected if \(\dim M\geq 5\) and \(\pi_1(M)\) contains an element that is not conjugate to its square. It appears to be unknown if there exist nontrivial finitely presented groups where every element is conjugate to its square. (Such a group would be perfect, non-residually-finite, and centerless as the author observes.) It is easy to see, as the author notes, that \(W(M):=Wh_1^+(\pi_1(M);\,\mathbb Z/2\oplus\pi_2(M))\) is nontrivial if \(M\) is non-simply-connected. (We note that a presentation of \(W(M)\) and the geometry of its generators are discussed in the last section of \textit{A. E. Hatcher}'s paper [Ann. Math. (2) 102, 101--137 (1975; Zbl 0305.57009)]; a recent discussion of the history of corrections of the latter paper can be found in \textit{F. Waldhausen}, the author and \textit{J. Rognes} [preprint, http://folk.uio.no/rognes/papers/plmf.pdf].) In his 1973 book with Wagoner, Hatcher claimed that \(\pi_0(\mathcal P(M))\simeq W(M)\oplus Wh_2(\pi_1(M))\) if \(\dim M\geq 6\) (which would have settled the aforementioned motivating question). This assertion was corrected by \textit{K. Igusa} [Lect. Notes Math. 1946, 104--172 (1984; Zbl 0546.57015)], who used an extension of Hatcher--Wagoner methods to show that there is an exact sequence \[ K_3(\mathbb Z[\pi_1(M)])@>\chi>>W(M)@>\phi>>\pi_0(\mathcal P(M))\to Wh_2(\pi_1(M))\to 0 \] when \(\dim M\geq 7\) and gave examples showing that \(\chi\) is nonzero in general. In order to detect nontrivial elements in the image of \(\phi\), the author uses Bökstedt's trace map \(A(M)\to\Omega^\infty S^\infty(\Lambda M)\), where \(\Lambda M\) is the free loop space and \(A(M)\) Walhausen's \(A\)-theory. The author also re-establishes Igusa's exact sequence by using Waldhausen's \(A\)-theory, and when \(\dim M\geq 9\) extends it further to the left: \[ \pi_1(\mathcal P(M))\to Wh_3(\pi_1(M))@>\chi>>W(M)@>\phi>>\pi_0(\mathcal P(M)) \to Wh_2(\pi_1(M))\to 0. \] In the case where \(\pi_2(M)=0\) this extension was also constructed by Igusa by different methods. The author's approach involves showing that the ``monomial'' homomorphism \(\pi_2^S(B\pi_+)\to K_2(\mathbb Z\pi)\) is split injective if \(\pi\) is finitely generated; he says that this result is well-known to the experts but might be not in the literature. In a side remark to the proof of this auxiliary result, the author also points out two corollaries of Waldhausen's results: (i) the homomorphism \(D_n:K_n(\mathbb Z\pi)\to H_n(\pi)\) is surjective for \(n=3\) (in particular, \(K_3(\mathbb Z[\mathbb Z/n])\) maps onto \(\mathbb Z/n\)), and (ii) an alternative proof of Dennis' result that \(D_n\) is surjective if \(H_i(\pi)=0\) for \(i<n\).
scientific article; zbMATH DE number 5806794
scientific article