Computational Topology: an introduction

Time:2:15 - 3:30pm, Tuesdays and Thursdays, Fall 2009.
Place:Gates 498
Instructor:Dmitriy Morozov (



Date Topic Notes
Tu, Sep 22 Introduction. Connected Components.

[EH09] Section I.1

On Wikipedia: Graph, Topological space, Connected space, Distjoint-set DS.
Th, Sep 24 Curves and Knots.

[EH09] Section I.2+3

On Wikipedia: Jordan Curve Theorem, Knot Theory, Links, Complexity.

Tu, Sep 29 Surfaces.

[EH09] Section II.1. See [M00] Sections 74,76,77 for rigorous treatment of polygonal schema, cutting pasting, and surface classification.

On Wikipedia: Surface, Cross-cap, Projective plane, Klein bottle.

Th, Oct 1 Fundamental Group. Homotopy.

HE's notes on Fundamental Group are not part of [EH09].

Afra Zomorodian's notes provide concise and clear review of the relevant topics in Group Theory. He also hosts English translation of Markov's paper on insolubility of homeomorphy.

Tu, Oct 6 Simplicial Complexes.

[EH09] Section III.1

On Wikipedia: Simplex, Simplicial complex, Abstract simplicial complex, Barycentric subdivision.

Th, Oct 8 Nerves.

[EH09] Sections III.2, III.3, III.4. [H01] Section 4.G, Bron-Kerbosch algorithm.

On Wikipedia: Nerve, Vietoris-Rips complex, Voronoi diagram, Delaunay triangulation.

Tu, Oct 13 Homology. Matrix Reduction.

[EH09] Sections IV.1, IV.2.

On Wikipedia: Homology, Reduced homology, Smith normal form.

Th, Oct 15 Relative Homology.

[EH09] Section IV.3.

On Wikipedia: Relative homology, Exact sequence.

Tu, Oct 20 Exact Sequences.

[EH09] Section IV.4.

On Wikipedia: Snake Lemma, Mayer-Vietoris sequence.

Th, Oct 22 Cohomology.

[EH09] Section V.1.

On Wikipedia: Cohomology.

Tu, Oct 27 Poincare Duality.

[EH09] Section V.2.

On Wikipedia: Poincare duality.

Th, Oct 29 Alexander Duality.

[EH09] Section V.4.

On Wikipedia: Alexander duality.

Tu, Nov 3 Smooth Generic Functions.

[EH09] Section VI.1. [M02].

On Wikipedia: Morse Theory.

Th, Nov 5 PL Functions.

[EH09] Section VI.3.

Tu, Nov 10 Reeb graphs.

[EH09] Section VI.4.

Th, Nov 12 Persistent Homology.

[EH09] Sections VII.1-2.

Tu, Nov 17 Stability.

[EH09] Sections VIII.1-2.

Th, Nov 19 Zigzag Persistence. [CdS08].
Tu, Dec 1 Extended Persistence

[EH09] Section VII.3.

Th, Dec 3 Levelset Zigzag.  


[EH09](1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18)

Herbert Edelsbrunner and John Harer. Computational Topology: an Introduction. AMS Press, 2009.


This book will not be available until January. However, it is a superset of course notes which can serve as a good supplement until the book is out.

[CdS08]Gunnar Carlsson and Vin de Silva. Zigzag Persistence. Manuscript, 2008.
[M02]Yukio Matsumoto. An Introduction to Morse Theory. AMS Press, 2002.
[H01]Allen Hatcher. Algebraic Topology. 2001.
[CLRS01]Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, Clifford Stein. Introduction to Algorithms. MIT Press, 2nd edition, 2001.
[M00]James R. Munkres. Topology. Prentice Hall, 2000.
[M84]James R. Munkres. Elements of Algebraic Topology. Perseus, 1984.

Leonidas Guibas and Dmitriy Morozov gratefully acknowledge the support to the Geometry Group provided by the Computer Forum during the 2009-10 academic year.