Invariants for homology 3-spheres /

Invariants for homology 3-spheres /

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Bibliographic Details
Author / Creator:Saveliev, Nikolai, 1966-
Imprint:Berlin ; New York : Springer, c2002.
Description:xii, 223 p. : ill. ; 25 cm.
Language:English
Series:Encyclopaedia of mathematical sciences, 0938-0396 ; v. 140. Low-dimensional topology ; 1
Encyclopaedia of mathematical sciences ; v. 140.
Encyclopaedia of mathematical sciences. Low-dimensional topology ; 1.
Subject:
Invariants.
Three-manifolds (Topology)
Invariants.
Three-manifolds (Topology)
Format: Print Book
URL for this record:http://pi.lib.uchicago.edu/1001/cat/bib/4840040
Hidden Bibliographic Details
Varying Form of Title:Invariants for homology three-spheres
ISBN:3540437967 (hd.bd.)
Notes:Includes bibliographical references (p. [205]-218) and index.
Table of Contents:
  • 1. Homology 3-Spheres
  • 1.1. Integral Homology 3 Spheres
  • 1.1.1. Homotopy 3-Spheres
  • 1.1.2. Poincaré Homology Sphere
  • 1.1.3. Brieskorn Homology Spheres
  • 1.1.4. Seifert Fibered Homology Spheres
  • 1.1.5. Dehn Surgery on Knots
  • 1.1.6. Surgery on Links
  • 1.1.7. Connected Sums and Splicing
  • 1.1.8. Splice Decomposition
  • 1.1.9. Plumbing
  • 1.1.10. Links of Singularities
  • 1.1.11. Mutations
  • 1.1.12. Branched Covers
  • 1.1.13. Heegaard Splittings of Homology Spheres
  • 1.2. Rational Homology Spheres
  • 1.2.1. Spherical Space Forms
  • 1.2.2. Dehn Surgery
  • 1.2.3. Seifert Fibered Manifolds
  • 1.2.4. Links of Singularities
  • 1.2.5. Branched Covers
  • 2. Rokhlin Invariant
  • 2.1. The Rokhlin Theorem
  • 2.2. Definition of the Rokhlin Invariant
  • 2.3. Properties of the Rokhlin Invariant
  • 2.3.1. Surgery Formula for the Rokhlin Invariant
  • 2.3.2. Surgery on Algebraically Split Links
  • 2.3.3. Splicing and Mutation
  • 2.3.4. Rokhlin Invariant of Branched Coverings
  • 2.3.5. Birman-Craggs Homomorphisms
  • 2.3.6. Homology Cobordism Invariance
  • 2.4. Seifert Fibered and Graph Homology Spheres
  • 2.4.1. The Algorithm
  • 2.4.2. The Formula
  • 3. Casson Invariant
  • 3.1. Definition of the Casson Invariant
  • 3.2. Construction of the Casson Invariant
  • 3.2.1. SU(2)-Represcntation Spaces
  • 3.2.2. The Intersection Theory
  • 3.2.3. Orientations
  • 3.2.4. Independence of Heegaard Splitting
  • 3.2.5. Casson Invariant for Knots and Property (1)
  • 3.2.6. The Difference Cycle
  • 3.2.7. Casson Invariant for Boundary Links and Property (2)
  • 3.2.8. Casson Invariant of a Trefoil and Property (0)
  • 3.3. Comments and Ramifications
  • 3.3.1. Pillowcase
  • 3.3.2. Perturbations
  • 3.3.3. The Connected Sum Formula
  • 3.3.4. The Integrality of ¿(¿)
  • 3.3.5. Casson Invariant of Algebraically Split Links
  • 3.4. Properties of the Casson Invariant
  • 3.4.1. Splicing Additivity
  • 3.4.2. Mutation Invariance
  • 3.4.3. Casson Invariant of Branched Coverings
  • 3.4.4. Casson Invariant of Fibered Knots
  • 3.4.5. Finite Type Invariants
  • 3.4.6. Further Properties of the Casson Invariant
  • 3.5. Seifert Fibered and Graph Homology Spheres
  • 3.5.1. Casson Invariant of ¿(p, q, r)
  • 3.5.2. Casson Invariant of ¿(a 1 , ..., a n )
  • 3.5.3. The Neumann-Wahl Conjecture
  • 3.6. Applications of the Casson Invariant
  • 3.6.1. Triangulating Topological 4-Manifolds
  • 3.6.2. Arnphicheiral Homology Spheres
  • 3.6.3. Property P for Knots
  • 4. Invariants of Walker and Lescop
  • 4.1. Definition of the Walker Invariant
  • 4.2. Construction of the Walker Invariant
  • 4.2.1. SU(2)-Representation Varieties
  • 4.2.2. The Intersection Theory
  • 4.2.3. The Surgery Formula
  • 4.2.4. Combinatorial Definition of the Walker Invariant
  • 4.3. The Lescop Invariant
  • 4.4. Properties of the Walker and Lescop Invariants
  • 4.4.1. The Gluing Formula
  • 4.4.2. Branched Covers
  • 4.4.3. Seifert Fibered Manifolds
  • 4.5. Casson Type Invariants from Other Lie Groups
  • 5. Casson Invariant and Gauge Theory
  • 5.1. Gauge Theory in Dimension 3
  • 5.2. Chern-Simons Function
  • 5.3. The Casson Invariant from Gauge Theory
  • 5.3.1. Morse Theory and Euler Characteristic
  • 5.3.2. Critical Points of cs and Spectral Flow
  • 5.3.3. Non-degenerate Case
  • 5.3.4. Perturbations
  • 5.3.5. Morse type Perturbations
  • 5.3.6. Casson Invariant and Seiberg-Witten Equations
  • 5.4. Casson-type Invariants of Knots
  • 5.4.1. Representation Varieties of Knot Groups
  • 5.4.2. The Invariants
  • 5.5. Equivariant Casson Invariant
  • 5.5.1. Equivariant Gauge Theory
  • 5.5.2. Definition of the Invariants
  • 5.5.3. Equivariant Casson and Knot Signatures
  • 5.5.4. Applications
  • 5.6. The SU(3) Casson Invariant
  • 5.6.1. Some SU(3)-Gauge Theory
  • 5.6.2. Definition of the Invariant
  • 5.6.3. Properties and Computations
  • 6. Instanton Floer Homology
  • 6.1. Gauge Theory in Dimension 4
  • 6.1.1. Gauge Theory on Closed 4-Manifolds
  • 6.1.2. Gauge Theory on Open 4-Manifolds
  • 6.1.3. Linear Analysis
  • 6.1.4. Non-linear Analysis
  • 6.2. Definition of the Floer Homology
  • 6.2.1. Review of the Morse Theory
  • 6.2.2. Floer Homology of Integral Homology Spheres
  • 6.2.3. Functoriality with Respect to Cobordisms
  • 6.3. Spectral Flow Formulas
  • 6.3.1. The Atiyah-Patodi-Singer Formula
  • 6.3.2. The Splitting Formula
  • 6.3.3. The Kirk-Klassen Formula
  • 6.4. Seifert Fibered and Graph Homology Spheres
  • 6.4.1. The Algorithm
  • 6.4.2. The Closed Form Formula
  • 6.4.3. Graph Homology Spheres
  • 6.5. Properties of the Floer Homology
  • 6.5.1. Orientation Reversal
  • 6.5.2. Floer Homology of Homology Handles
  • 6.5.3. The Floer Exact Triangle
  • 6.5.4. Special Boundary Maps
  • 6.5.5. The u-map in Floer Homology
  • 6.5.6. Integer Graded Floer Homology
  • 6.5.7. Floer Homology of Connected Sums
  • 6.5.8. Functoriality with Respect to Diffeornorphisms
  • 6.5.9. Mutation Invariance
  • 6.6. Floer Homology in Donaldson Theory
  • 6.6.1. Donaldson Invariants of Closed 4-Manifolds
  • 6.6.2. Relative Donaldson Polynomials
  • 6.6.3. The Gluing Formula
  • 6.7. Extending Floer Homology
  • 6.7.1. Equivariant Floer Homology
  • 6.7.2. Fukaya-Floer Homology
  • 6.7.3. Floer's Category and Functor
  • 6.7.4. The Atiyah-Floer Conjecture
  • 6.7.5. Floer Homology of Knots
  • 6.7.6. Seiberg-Witten Floer Homology
  • 7. The Homology Cobordism Group
  • 7.1. Homology Cobordisms
  • 7.1.1. Mazur Homology Spheres
  • 7.1.2. Knot Cobordisms and Homology Cobordisms
  • 7.1.3. Ribbon Concordances and Ribbon Knots
  • 7.1.4. Branched Coverings
  • 7.2. The Structure of \Theta_{{\op Z}}^3
  • 7.2.1. A Classical Approach to \Theta_{{\op Z}}^3
  • 7.2.2. Infinite Order Elements in \Theta_{{\op Z}}^3
  • 7.2.3. The \bar \mu -Invariant
  • 7.2.4. The Fukumoto-Furuta Invariants
  • 7.2.5. The Group \Theta_{{\op Z}}^3 is Infinitely Generated
  • 7.3. Applications of the Homology Cobordism Group
  • 7.3.1. Triangulating Topological Manifolds
  • 7.3.2. Knot Concordance Group
  • 7.3.3. PL-discs in Contractible 4-Manifolds
  • 7.3.4. Constructing Smooth Manifolds
  • References
  • Index

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