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Peking Summer School (July 2005)

Class Photo: The Summer School Class in Topology met six hours each week and had over 70 students.
Jaco with host Professor Wang Shicheng at the Mathematics Institute in Peking University During lectures on the homeomorphism problem for 3-manifolds

TITLE: "The Homeomorphism Problem: Classification of 3--Manifolds"

Given two 3-manifolds M and N, the Homeomorphism Problem is to determine if M and N are homeomorphic (topologically equivalent). In dimension three, this is equivalent to the problem of classifying three-manifolds and is one of the most important problems in low-dimensional topology. This series of lectures outline the solution of the Homeomorphism Problem for 3-manifolds up to the claim by G.Perelman of W.Thurston's Geometrization Conjecture.

Lecture 1: Introduction to the Homeomorphism Problem.

This lecture introduces the basic structure theorems for 3-manifolds (Prime Decomposition, JSJ Decomposition and Thurston's Geometrization Conjecture) and outlines the solution to the Homeomorphism Problem (and Classification) for 3- manifolds.

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Lecture 2: On Classifying Manifolds.

This lecture gives a slightly new approach to the classification of 2-manifolds and a new proof of the topological invariance of Euler characteristic for 2-manifolds; we discuss difficulties of extending these methods to the classification of 3-manifolds; and show the impossibility of classifying n-manifolds for larger than 3.

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Lectures 3 and 4: Presentations of 3-manifolds.

These two lectures explore various presentations of 3-manifolds via triangulations, cell-decompositions, handle-decompositions, Heegaard splittings Heegaard diagrams, and knot and link projections. Algorithms are given that transform each of these presentations into a triangulation.

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Lecture 5: Normal Surface Theory: A Brief Survey.

This lecture gives a basic introduction to the theory of normal surfaces and establish some of the fundamental existence theorems.

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Lecture 6: Normal Surface Theory: Parameterizations and Algorithms.

This lecture explores some of the applications ofnormal surfaces to decision problems in 3-manifold topology.

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Lecture 7: Prime Decomposition of 3-manifolds.

This lecture provides the existence and uniqueness theorems of H.Kneser and J.Milnor for the prime decomposition of a 3-manifolds.

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Lecture 8: An Algorithm to Construct the Prime Decomposition of a 3-manifold.

This lecture uses the new methods of 0-efficient triangulations to give an algorithm for constructing the prime decomposition of a 3-manifold.

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Lecture 9: JSJ Decomposition of 3-manifolds.

This lectures gives a brief introduction to Seifert fibered 3-manifolds and provides the existence and uniqueness theorem of Jaco-Shalen and Johannson for the JSJ Decomposition of a 3-manifold.

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Lecture 10: An Algorithm to Construct the JSJ Decomposition of a 3-manifold.

An algorithm is given for constructing the JSJ-decomposition of a 3-manifold and deriving the Seifert invariants of the Characteristic submanifold.

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Lecture 11: The Geometrization Conjecture.

This lecture discusses the Geometrization Conjecture, the eight locally homogeneous geometries for 3-manifolds and Perelman's Claim of a solution to the Geometrization Conjecture and its implications.


Lecture 12: The Homeomorphism Problem for Haken and Hyperbolic 3-manifolds.

This lecture outlines the solution of the Homeomorphism Problem for Haken 3-manifolds and discusses Sela's solution of the Homeomorphism Problem for closed hyperbolic 3-manifolds.