Unitary nilpotent groups and the stability of pseudoisotopies
For a manifold \(M\), the difference between an isotopy and a pseudoisotopy of \(M\) is measured by the 0-dimensional homotopy grup \(\pi_ 0 (C(M))\) of the concordance space \(C(M)\), called also the pseudoisotopy space of \(M\). If \(\dim M \geq 5\), one can provide a complete description of \(\pi_ 0 (C(M))\) in terms of higher algebraic \(K\)-theory invariants [cf. \textit{A. E. Hatcher}, Proc. Symp. Pure Math., Vol. 32, Part 1, 3-21 (1978; Zbl 0406.57031), \textit{A. E. Hatcher} and \textit{J. B. Wagoner}, Astérisque 6, 8-238 (1973; Zbl 0274.57010), \textit{A. E. Hatcher}, ibid., 239-275 (1973; Zbl 0274.57011) and \textit{K. Igusa}, Lect. Notes Math. 1946, 104-172 (1984; Zbl 0546.57015)]. The main results of the paper under review show that if \(\dim M = 3\), the Hatcher-Wagoner invariants do not necessarily detect all elements of \(\pi_ 0 (C(M))\) and certain nonzero invariants are realized by elements of \(\pi_ 0 (C(N))\) for certain 3-manifolds \(N\). The paper shows the lack of the stability property and the failure of the product formula for pseudoisotopies on 3-manifolds. The results of the paper provide also further evidence for the fact that there exist pseudoisotopic, but not isotopic, homeomorphisms on some closed reducible 3-manifolds, as well as on irreducible 3-manifolds with nonempty boundary.
0-dimensional homotopy group of the concordance space
pseudoisotopic, but not isotopic, homeomorphisms
scientific article; zbMATH DE number 558990
scientific article