Casson's invariant for homology 3-spheres and the mapping class group
type: Item
instance of: scholarly article
title: Casson's invariant for homology 3-spheres and the mapping class group
zbMATH Open document ID: 0623.57006
author: Shigeyuki Morita
published in: Proceedings of the Japan Academy. Series A
publication date: 1986-01-01T00:00:00Z
review text: In 1978 \textit{J. S. Birman} and \textit{R. Craggs} [Trans. Am. Math. Soc. 237, 283--309 (1978; Zbl 0383.57006)] introduced \(\mathbb{Z}/2\)-invariants of homeomorphisms of surfaces, making use of the Rokhlin invariants of homology 3-spheres. The paper under review is considered with a refinement of Birman-Craggs invariants based on the Casson integer invariant \(\lambda\) of oriented homology 3-spheres. Let \(S\) be the closed orientable surface of genus \(\geq 2\), and let \(S_0=S\setminus \operatorname{Int} D\) where \(D\) is a disk in \(S\). Let \(E\) be the set of all embeddings \(S\hookrightarrow S^3\). Let \(G_1,G_2,\ldots\), \(G_{n+1}=[G_n,G_1],\ldots\) be the lower central series of \(\pi_1(S_0)=G_1\). Let \(M_n\) be the subgroup of the mapping class group of \(S_0\) consisting of the elements which act trivially on \(G_1/G_n\). The author defines a mapping \(\Lambda: M_2\to \operatorname{Map}(E, \mathbb{Z})\): If \(\phi \in M_2\) and \(f\in E\) then \(\Lambda (\phi)(f)=\lambda (W_{\phi})\), where \(W_{\phi}\) is the oriented homology 3-sphere obtained by cutting \(S^3\) along \(f(S)\) and regluing back the resulting two pieces by the extension of \(\phi\) to \(S\). Several theorems concerned with computation and properties of \(\Lambda\) are announced. In particular, the value \(\Lambda(\phi)\) is stated to depend only on the class of \(\phi\) in \(M_2/M_5\). This implies that it is possible to define the Casson invariant of homology 3-spheres in terms of the action of the pasting maps of Heegaard decompositions on the fifth nilpotent quotient of the fundamental group of the Heegaard surface.
Mathematics Subject Classification ID: 57K30
Mathematics Subject Classification ID: 57K20
Mathematics Subject Classification ID: 57R50
Mathematics Subject Classification ID: 57K31
zbMATH DE Number: 4011436
zbMATH Keywords: mapping class group
zbMATH Keywords: Rochlin's invariant
zbMATH Keywords: Torelli group
zbMATH Keywords: Casson integer invariant of oriented homology 3-spheres
zbMATH Keywords: homeomorphisms of surfaces
zbMATH Keywords: Birman-Craggs invariants
zbMATH Keywords: fundamental group of the Heegaard surface
reviewed by: Vladimir Turaev
MaRDI profile type: Publication
full work available at URL: pjaa.62.402
OpenAlex ID: W2002708046
cites work: The μ-Invariant of 3-Manifolds and Certain Structural Properties of the Group of Homeomorphisms of a Closed, Oriented 2-Manifold
cites work: Quadratic Forms and the Birman-Craggs Homomorphisms
cites work: An abelian quotient of the mapping class group \(\mathfrak S\)
cites work: A survey of the Torelli group
cites work: The structure of the Torelli group. II: A characterization of the group generated by twists on bounding curves
cites work: Q5513066
DOI: 10.3792/PJAA.62.402
description: scientific article; zbMATH DE number 4011436