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Algebraic Topology I

Columbia University, Fall semester, 2008

  • Mondays and Wednesdays, 11 am - 12:15 pm, room 507 (Mathematics Building)

  • Instructor: Tim Perutz (Ritt Assistant Professor)

  • Office hours: Mondays and Tuesdays, 1:30-2:30 pm, 413 Math.

  • TA: Jonathan Bloom.

Outline

Classical algebraic topology is a theory relevant to mathematicians in many fields: there are direct connections to geometric and differential topology, algebraic and differential geometry, global analysis, mathematical physics, group theory, homological algebra and category theory; and points of contact with other areas including number theory. The core of the subject - the theory of homology and cohomology - began with Poincaré in the 1890s, and finally crystallized with Eilenberg and MacLane in the early 1950s. It is a self-contained and economical story, and I hope you'll agree that it's an enjoyable and beautiful one too. Dylan Thurston's course, Algebraic Topology II, in the spring semester, will take the subject further, developing homotopy theory and its relations with cohomology.

We shall begin with the fundamental group, and go on to homology and cohomology. I expect you to know basic topological terms such as "Hausdorff topological space" and algebraic ones such as "normal subgroup", but I will not assume any acquaintance with algebraic topology or homological algebra.

Textbooks

Algebraic topology textbooks fall on a spectrum: at one extreme there are books that emphasize the geometric aspects of the subject, often working by example. Perhaps the best of these is Allen Hatcher's Algebraic topology: we will work through large chunks of his chapter 0-3. At the other extreme, there are books that emphasize the formal aspects, developing a streamlined language for converting topology to algebra, as for example Peter May's accurately titled A concise course in algebraic topology. This viewpoint is also valuable, and I will occasionally draw on May's exposition. However your priority should be to develop topological intuition through examples rather than to learn how to prove things in great generality.

References:

Allen Hatcher, Algebraic topology, Cambridge U.P, 2002. Freely available online here.

J. Peter May, A concise course in algebraic topology, Chicago Lectures in Mathematics, University of Chicago Press, 1999. Freely available online here.

Neither book is expensive, so do consider buying the paper version(s). Other classics include Spanier's Algebraic topology, Bott and Tu's Differential forms in algebraic topology, and Milnor's Topology from the differentiable viewpoint.

Assessment

There will be approximately 8 homework assignments, collected in the Wednesday class. There will also be a final exam.

Homework assignments

First homework: Exercises 1.1-1.7 from lecture 1. Download the notes (containing the exercises) from the link at the bottom of this page. Due at the beginning of class, Wednesday September 10.

Syllabus

  • The fundamental group (7 lectures)

    Spaces and their homotopy equivalence. The fundamental group. (Hatcher 1.1)

    Fundamental group of the circle; FTA and Brouwer. (Hatcher 1.1)

    Van Kampen in theory and practice. (Hatcher 1.2.)

    Covering spaces. (Hatcher 1.3)

  • Singular homology (9 lectures)

    Definition of singular homology. Relation of H_0 to pi_0 and of H_1 to pi_1. Homotopy invariance. (Hatcher 2.1, 2.A.)

    Chain complexes and chain homotopy. Short and long exact sequences. (Hatcher 2.1)

    Mayer-Vietoris. Homology of spheres. (Hatcher 2.2.)

    Relative homology, exact sequence of the pair. Excision. (Hatcher 2.1-2.2.)

    Vanishing theorems for singular homology of manifolds. Orientations and fundamental classes. (Hatcher 3.3)

    Tor and universal coefficients. (Hatcher 3.A.)

  • Cellular homology (3 lectures)

    CW complexes and their topological properties. (Hatcher 0, 2.2.)

    Deduction of cellular from singular homology. Uniqueness of Eilenberg-Steenrod theories. (Hatcher 2.2, 2.3.)

    Computations of in cellular homology. (Hatcher 2.2.)

  • Product structures (6 lectures)

    Singular and cellular cohomology. Ext and dual universal coefficients. (Hatcher 3.1.)

    Properties of the cup product. The Poincaré pairing. Projective spaces. Applications. (Hatcher 3.2, 3.3.)

    Cup and cap products defined. (Hatcher 3.2.)

    Proof of Poincaré duality. (Hatcher 3.3)

Lecture notes

These notes have little or no claim to originality and are intended just for use in my course. They are based on other sources, mainly the books of Hatcher and May mentioned above.

Help with finding errors is appreciated.

Complete course