Lectures on the Topology of 3-Manifolds: An Introduction to the Casson Invariant

ISBN-13: 9783110250350 / Angielski / Twarda / 2011 / 218 str.

Nikolai Saveliev

Opis

This textbook, now in its second revised and extended edition, introduces the topology of 3- and 4-dimensional manifolds. It also considers new developments especially related to the Heegaard Floer and contact homology. The book is accessible to graduate students in mathematics and theoretical physics familiar with some elementary algebraic topology, including the fundamental group, basic homology theory, and Poincaré duality on manifolds.

Spis treści

Preface
Introduction
Glossary

1 Heegaard splittings
1.1 Introduction
1.2 Existence of Heegaard splittings
1.3 Stable equivalence of Heegaard splittings
1.4 The mapping class group
1.5 Manifolds of Heegaard genus ≤ 1
1.6 Seifert manifolds
1.7 Heegaard diagrams

2 Dehn surgery
2.1 Knots and links in 3-manifolds
2.2 Surgery on links in S³
2.3 Surgery description of lens spaces and Seifert manifolds
2.4 Surgery and 4-manifolds

3 Kirby calculus
3.1 The linking number
3.2 Kirby moves
3.3 The linking matrix
3.4 Reversing orientation

4 Even surgeries

5 Review of 4-manifolds
5.1 Definition of the intersection form
5.2 The unimodular integral forms
5.3 Four-manifolds and intersection forms

6 Four-manifolds with boundary
6.1 The intersection form
6.2 Homology spheres via surgery on knots
6.3 Seifert homology spheres
6.4 The Rohlin invariant

7 Invariants of knots and links
7.1 Seifert surfaces
7.2 Seifert matrices
7.3 The Alexander polynomial
7.4 Other invariants from Seifert surfaces
7.5 Knots in homology spheres
7.6 Boundary links and the Alexander polynomial

8 Fibered knots
8.1 The definition of a fibered knot
8.2 The monodromy
8.3 More about torus knots
8.4 Joins
8.5 The monodromy of torus knots
8.6 Open book decompositions

9 The Arf-invariant
9.1 The Arf-invariant of a quadratic form
9.2 The Arf-invariant of a knot

10 Rohlin’s theorem
10.1 Characteristic surfaces
10.2 The definition of ˜q
10.3 Representing homology classes by surfaces

11 The Rohlin invariant
11.1 Definition of the Rohlin invariant
11.2 The Rohlin invariant of Seifert spheres
11.3 A surgery formula for the Rohlin invariant
11.4 The homology cobordism group

12 The Casson invariant

13 The group SU(2)

14 Representation spaces
14.1 The topology of representation spaces
14.2 Irreducible representations
14.3 Representations of free groups
14.4 Representations of surface groups
14.5 Representations for Seifert homology spheres

15 The local properties of representation spaces

16 Casson’s invariant for Heegaard splittings
16.1 The intersection product
16.2 The orientations
16.3 Independence of Heegaard splitting

17 Casson’s invariant for knots
17.1 Preferred Heegaard splittings
17.2 The Casson invariant for knots
17.3 The difference cycle
17.4 The Casson invariant for boundary links
17.5 The Casson invariant of a trefoil

18 An application of the Casson invariant
18.1 Triangulating 4-manifolds
18.2 Higher-dimensional manifolds

19 The Casson invariant of Seifert manifolds
19.1 The space R(p; q; r)
19.2 Calculation of the Casson invariant

Conclusion
Bibliography
Index

O autorze

Nikolai Saveliev, University of Miami, Florida, USA.