Algebraic Topology

John Baez, Mike Stay, Christopher Walker

Winter 2007

Here are some notes for an introductory course on algebraic topology. The lectures are by John Baez, except for classes 2-4, which were taught by Derek Wise. The lecture notes are by Mike Stay.

Homework assigned each week was due on Friday of the next week. You can read answers to these homework problems, written by Christopher Walker.

The course used this book:

James Munkres, Topology, 2nd edition, Prentice Hall, 1999.

So, theorem numbers match those in this book whenever possible, and it's best to read these notes along with the book. We deviate from Munkres at various points. We skip many sections, and we put more emphasis on concepts from category theory, especially near the end of the course.

But, the star of the show is π₁ — the fundamental group!

Class 1 (Jan. 5) - Sketch of how we'll use the fundamental group to prove there's no retraction from the disk to the circle.

Class 2 (Jan. 8) - Definition of path homotopy, fundamental group. Proof that π₁ is a functor.

Class 3 (Jan. 10) - Change of basepoint. Simply-connected spaces. Covering spaces.

Class 4 (Jan. 12) - Covering maps. Liftings.
- Homework due January 12.

Class 5 (Jan. 17) - Goal: computing the fundamental group of the circle, S¹. The lifting map (part 1).

Class 6 (Jan. 19) - The lifting map (part 2).
- Homework due January 19.

Class 7 (Jan. 22) - The lifting map (part 3). Proof that the fundamental group of S¹ is Z. Consequence: there is no retraction from the disc to the circle.

Class 8 (Jan. 24) - More consequences: the identity map from the circle to itself is not nulhomotopic. Any nonvanishing vector field on the disc points directly outwards somewhere on the boundary. The Brouwer Fixed Point Theorem: every map from the disc to itself has a fixed point.

Class 9 (Jan. 26) - Generalizing everything we've done so far from the circle to higher-dimensional spheres, using the homotopy groups π_n.
- Homework due January 26.

Class 10 (Jan. 29) - The Fundamental Theorem of Algebra.

Class 11 (Jan. 31) - The big picture: each branch of mathematics studies some category. Definition of category, functor.

Class 12 (Feb. 2) - Definition of homotopy equivalence, homotopy type. Pointed spaces, pointed maps, and pointed homotopies.
- Homework due February 2.

Class 13 (Feb. 5) - Homotopy equivalent spaces have isomorphic fundamental groups. Examples of homotopy equivalences. Deformation retracts.

Class 14 (Feb. 7) - The Baby Seifert-van Kampen Theorem (part 1).

Class 15 (Feb. 9) - The Baby Seifert-van Kampen Theorem (part 2). The n-sphere Sⁿ is simply connected for n > 1. The real projective space RPⁿ.
- Homework due February 9.

Class 16 (Feb. 10) - The fundamental group of RPⁿ is Z/2 for n > 1.

Class 17 (Feb. 14) - The Seifert-van Kampen Theorem. Pushouts. Examples of pushouts.

Class 18 (Feb. 16) - The free group on 2 generators as a pushout. The fundamental group of the figure 8 is the free group on 2 generators. Pushouts are unique up to isomorphism. The free product of groups as a pushout.
- Homework due February 16.

Class 19 (Feb. 21) - The fundamental group of a bouquet of circles. General pushouts of groups.

Class 20 (Feb. 23) - Practical version of the Seifert-van Kampen Theorem. Computing the fundamental group of the torus using this theorem. The fundamental group of the two-holed torus. The fundamental group of the dunce cap.
- Homework due February 23.

Class 21 (Feb. 26) - Munkres' version of the Seifert-van Kampen Theorem: sketch of the proof.

Class 22 (Feb. 28) - From Munkres' version of the Seifert-van Kampen Theorem to the general case. The fundamental group of the Klein bottle. The infinite dihedral group.

Class 23 (Mar. 2) - Limits and colimits. Pullbacks and pushouts. Products and coproducts. Examples.
- Homework due March 2.

Class 24 (Mar. 5) - The coproduct of pointed spaces — also known as the wedge product. The functor π₁ preserves coproducts for well-pointed spaces.

Class 25 (Mar. 7) - The product of pointed spaces — also known as the Cartesian product. The functor π₁ preserves products for all pointed spaces. The fundamental group of the n-torus Tⁿ.

Class 26 (Mar. 9) - Eilenberg-Mac Lane spaces. Given a group G, the Eilenberg-Mac Lane space K(G,1) has G as its fundamental group, and vanishing higher homotopy groups.
- Homework due March 9.
Class 27 (Mar. 12) - Groupoids. The fundamental groupoid. The Better Seifert-van Kampen Theorem.

Class 28 (Mar. 14) - The classifying space of a groupoid.

Class 29 (Mar. 16) - Review. Are various subspaces retracts? Are they deformation retracts?
- Homework due March 16.

Try doing the review problems for the final exam, and then try doing the final exam.

If for some reason you want the LaTeX files of Christopher Walker's homework answers, the review problems, or the final, they're here. However, all rights to these belong to us.