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 24, 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 π_{1} — 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 π_{1} is a functor.

Class 3 (Jan. 10)  Change of basepoint.
Simplyconnected spaces. Covering spaces.

Class 4 (Jan. 12)  Covering maps. Liftings.

Class 5 (Jan. 17)  Goal: computing the fundamental
group of the circle, S^{1}. The lifting map (part 1).

Class 6 (Jan. 19)  The lifting map (part 2).

Class 7 (Jan. 22)  The lifting map (part 3).
Proof that the fundamental group of S^{1} 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 higherdimensional spheres,
using the homotopy groups π_{n}.

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.

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

Class 14 (Feb. 7)  The Baby Seifertvan Kampen
Theorem (part 1).

Class 15 (Feb. 9)  The Baby Seifertvan Kampen
Theorem (part 2). The nsphere S^{n} is simply connected for n > 1.
The real projective space RP^{n}.

Class 16 (Feb. 10)  The fundamental group of
RP^{n} is Z/2 for n > 1.

Class 17 (Feb. 14)  The Seifertvan 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.

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

Class 20 (Feb. 23)  Practical version of
the Seifertvan Kampen Theorem. Computing the fundamental group of
the torus using this theorem. The fundamental group of the twoholed torus.
The fundamental group of the dunce cap.

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

Class 22 (Feb. 28)  From Munkres' version
of the Seifertvan 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.

Class 24 (Mar. 5)  The coproduct of pointed
spaces — also known as the wedge product. The functor
π_{1} preserves coproducts for wellpointed spaces.

Class 25 (Mar. 7)  The product of pointed
spaces — also known as the Cartesian product. The functor
π_{1} preserves products for all pointed spaces. The
fundamental group of the ntorus T^{n}.

Class 26 (Mar. 9)  EilenbergMac Lane spaces.
Given a group G, the EilenbergMac Lane space K(G,1) has G as its
fundamental group, and vanishing higher homotopy groups.

Class 27 (Mar. 12)  Groupoids.
The fundamental groupoid. The Better Seifertvan 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?
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.
© 2007 John Baez, Mike Stay, Christopher Walker
baez@math.removethis.ucr.andthis.edu