Unitary nilpotent groups and the stability of pseudoisotopies
type: Item
instance of: scholarly article
title: Unitary nilpotent groups and the stability of pseudoisotopies
zbMATH Open document ID: 0804.57009
DOI: 10.1215/S0012-7094-93-07133-5
author: Reinhard Schultz
author: Slawomir Kwasik
published in: Duke Mathematical Journal
publication date: 1995-01-24T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 57N37
Mathematics Subject Classification ID: 57Q60
Mathematics Subject Classification ID: 57N10
Mathematics Subject Classification ID: 57R52
zbMATH DE Number: 558990
zbMATH Keywords: 0-dimensional homotopy group of the concordance space
zbMATH Keywords: isotopy
zbMATH Keywords: pseudoisotopy
zbMATH Keywords: pseudoisotopy space
zbMATH Keywords: Hatcher-Wagoner invariants
zbMATH Keywords: 3- manifolds
zbMATH Keywords: stability property
zbMATH Keywords: product formula for pseudoisotopies
zbMATH Keywords: pseudoisotopic, but not isotopic, homeomorphisms
reviewed by: Krzysztof Pawałowski
MaRDI profile type: Publication
cites work: The concordance-homotopy groups of geometric automorphism groups
cites work: Geometric Transfer and the Homotopy Type of the Automorphism Groups of a Manifold
cites work: Groups of automorphisms of manifolds. With an appendix by E. Peders en
cites work: Splitting obstructions for Hermitian forms and manifolds with 𝑍₂⊂𝜋₁
cites work: Unitary nilpotent groups and Hermitian $K$-theory. I
cites work: A splitting theorem for manifolds
cites work: La stratification naturelle des espaces de fonctions différentiables réelles et le théorème de la pseudo-isotopie
cites work: K2 (ZΦ) for Φ a Group of Order Two or Three
cites work: Q4040346
cites work: Homotopy is not isotopy for homeomorphisms of 3-manifolds
cites work: Q3910672
cites work: Higher simple homotopy theory
cites work: Q4192703
cites work: Q3903734
cites work: On the Diffeomorphism Group of S 1 × S 2
cites work: A proof of the Smale conjecture, \(Diff(S^ 3)\simeq O(4)\)
cites work: Q5182878
cites work: Q4041757
cites work: Q3319432
cites work: Parametrized surgery and isotopy
cites work: Q3336394
cites work: The stability theorem for smooth pseudoisotopies
cites work: Homotopy of spaces of diffeomorphisms of some three-dimensional manifolds
cites work: Q4746510
cites work: Pseudo-Isotopies of Irreducible 3-Manifolds
cites work: Low-dimensional concordances, Whitney towers and isotopies
cites work: Pseudo-isotopies of 3-manifolds
cites work: Lower bounds for \(K_ 2^{top}({\hat {\mathbb{Z}}}_ p\pi)\) and \(K_ 2({\mathbb{Z}}\pi)\)
cites work: \(K_ 2\) of p-adic group rings of abelian p-groups
cites work: \(SK_ 1\) of finite group rings. V
cites work: Pseudo-isotopies et isotopies en dimension quatre dans la catégorie topologique. (Pseudo-isotopies and isotopies of dimension four in the topological category)
cites work: Isotopy of 4-manifolds
cites work: Homotopy decompositions of equivariant function spaces. I: Spaces of principal bundle maps
cites work: Q5624419
cites work: Q3048986
cites work: Algebraic K-theory of generalized free products. I
cites work: Algebraic K-theory of generalized free products. II
cites work: Q3947813
full work available at URL: s0012-7094-93-07133-5
OpenAlex ID: W2073712579
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description: scientific article; zbMATH DE number 558990