A new definition of the Casson invariant
type: Item
instance of: scholarly article
title: A new definition of the Casson invariant
zbMATH Open document ID: 0996.57007
DOI: 10.1016/S1631-073X(02)02208-2
author: Bernard Perron
published in: Comptes Rendus. Mathématique. Académie des Sciences, Paris
publication date: 2002-11-02T00:00:00Z
review text: In a previous paper, following work of \textit{S. Morita} [Topology 28, No. 3, 305--323 (1989; Zbl 0684.57008)], the author defined an invariant \(\delta (f)\) for any \(f\) in the mapping class group of a compact oriented surface with one boundary component, and showed, using the surgery formula for Casson's invariant of homology 3-spheres, that if \(f\) is an element of a certain subgroup of the Torelli group (the subgroup of the mapping class group acting trivially on homology), then \(\delta (f)\) coincides with the Casson invariant of the homology 3-sphere \(M_f\) obtained by gluing two handlebodies along their boundaries using \(f\) (also, every homology 3-sphere is obtained in this way). In the present paper, it is shown directly that \(\delta (f)\) depends only on \(M_f\) (i.e. without reference to the Casson invariant), by showing that \(\delta (f)\) is invariant under the two Reidemeister-Singer operations for Heegaard splittings (stabilization and equivalence). This requires some ``lengthy, but elementary'' computations; on the other hand, the surgery formula, which is a difficult point in Casson's approach, follows almost immediately from the definition of \(\delta (f)\).
Mathematics Subject Classification ID: 57M27
zbMATH DE Number: 1744378
zbMATH Keywords: homology 3-sphere
zbMATH Keywords: mapping class group of a surface
reviewed by: Bruno Zimmermann
MaRDI profile type: Publication
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cites work: Casson's invariant for homology 3-spheres and characteristic classes of surface bundles. I
cites work: Introduction to Algebraic K-Theory. (AM-72)
cites work: Q4107635
cites work: Lectures on the topology of 3-manifolds. An introduction to the Casson invariant
cites work: On Homeomorphisms of a 3-Dimensional Handlebody
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description: scientific article; zbMATH DE number 1744378