Casson's invariant for homology 3-spheres and characteristic classes of surface bundles
type: Item
instance of: scholarly article
title: Casson's invariant for homology 3-spheres and characteristic classes of surface bundles
zbMATH Open document ID: 0641.57004
author: Shigeyuki Morita
published in: Proceedings of the Japan Academy. Series A
publication date: 1987-01-01T00:00:00Z
review text: Let \(H_ g\) be a 3-dimensional handle-body of genus g and \(S_ g=\partial H_ g\), a closed surface of genus g. The Torelli group \(T_{g,1}\) of \(S_ g\), relative an embedded disc D 2 in \(S_ g\), is the group of mapping classes of \(S_ g\) relative D 2 inducing the identity on homology. Given an element \(\phi\) in \(T_{g,1}\), there is a canonical way to glue together two copies of \(H_ g\) along their boundaries to produce (a Heegaard splitting of) a homology 3-sphere M(\(\phi)\), so the Casson invariant \(\lambda\) (M(\(\phi)\))\(\in {\mathbb{Z}}\) is defined and gives a map \(\lambda\) : \(T_{g,1}\to {\mathbb{Z}}\). ``The purpose of the present note is to announce our result concerning the map \(\lambda\). Briefly speaking we have shown that the Casson invariant is a kind of secondary invariant associated with the characteristic classes of surface bundles introduced in several papers of the author. As a result we have obtained an alternative description of \(\lambda\).''
Mathematics Subject Classification ID: 57N10
Mathematics Subject Classification ID: 57R20
Mathematics Subject Classification ID: 55R10
zbMATH DE Number: 4044759
zbMATH Keywords: 3-dimensional handle-body
zbMATH Keywords: Torelli group
zbMATH Keywords: group of mapping classes
zbMATH Keywords: homology 3-sphere
zbMATH Keywords: Casson invariant
zbMATH Keywords: characteristic classes of surface bundles
reviewed by: Bruno Zimmermann
MaRDI profile type: Publication
full work available at URL: pjaa.63.229
OpenAlex ID: W2093455531
cites work: On Siegel's modular group
cites work: Braids, Links, and Mapping Class Groups. (AM-82)
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: 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. III: The abelianization of \({\mathcal S}\)
cites work: Characteristic classes of surface bundles
cites work: Families of Jacobian manifolds and characteristic classes of surface bundles. I
cites work: Casson's invariant for homology 3-spheres and the mapping class group
cites work: On Homeomorphisms of a 3-Dimensional Handlebody
DOI: 10.3792/PJAA.63.229
description: scientific article; zbMATH DE number 4044759