Basic Topology of 3-Manifolds

Some time in the 1980’s I started writing a book on 3-manifolds, but got sidetracked on the algebraic topology books described above. The little that exists of the 3-manifolds book is rather crude and unpolished, and doesn’t cover a lot of material, but it does contain a few things that aren’t readily available elsewhere, like the elementary form of the Jaco-Shalen/Johannson torus decomposition theorem.

Η Κ-Θεωρία ανήκει στην κλάση των λεγόμενων γενικευμένων (ή εξωτικών) θεωριών ομολογίας και η χρησιμότητά της στα περίπου 50 χρόνια ζωής της έχει αποδειχθεί ήδη πολύ μεγάλη σε πολλούς τομείς των μαθηματικών όπως στην τοπολογία, την ολική ανάλυση, τη θεωρία αριθμών, την άλγεβρα, την γεωμετρία κλπ. Ενδεικτικά αναφέρουμε ότι αποτελεί το ένα από τα δύο βασικά συστατικά των περίφημων Θεωρημάτων Δείκτου που αποτελούν αναμφίβολα το πιο κομβικό αποτέλεσμα στα μαθηματικά του δεύτερου μισού 20ου αιώνα αφού ενώνει τρεις διαφορετικούς τομείς των μαθηματικών: την Ανάλυση, την Γεωμετρία (Τοπολογία) και την Άλγεβρα.

This is a shortened version of introduction to book “Topological Properties of Real Plane Algebraic Curves” by V.M.Kharlamov, V.A.Rokhlin and O.Y.Viro. This book is still under preparation. We have just taken away all the references to the main body of the book.

The aim of this course is to understand the role of differential forms in algebraic topology, and to use them to prove a number of important theorems about manifolds, to develop a grounding in Morse theory, to learn the basics of vector bundles and their spaces of sections, to introduce the theory of characteristic cohomology classes cononically associated with vector bundles.

The aim of the course is to give an exposition of some of these results which are not widely available. In particular, we will show that 3-manifolds with essential laminations satisfy the “weak geometrization conjecture”, and we will discuss the extent to which the tools of taut foliation theory can be generalized to a “coarse” context.

• Chapter 1: Fundamental Concepts
  • Describing Topological Spaces
  • Closed sets and Closure
  • Continuity and Homeomorphism
  • Additional Observations.

The book consists of two parts. General topology is the subject of the first one. The second part is an introduction to algebraic topology via its most classical and elementary segment which emerges from the notions of fundamental group and covering space.

The book consists of two parts. General topology is the subject of the first one. The second part is an introduction to algebraic topology via its most classical and elementary segment which emerges from the notions of fundamental group and covering space.