Math 215a: Algebraic topology
UC Berkeley, Fall 2007
- (12/12) Here are Ka Choi's notes on the
lectures. Many thanks to him for taking these notes and letting
me post them here.
- (8/29) I will be away at a conference next week. Professor Jones
has kindly agreed to give the lecture on Wednesday 9/5. I think he
will discuss applications of the fundamental group to knot theory.
This should be a real treat. The lecture on 9/7 is cancelled; I will
try to make it up later in the semester at a mutually convenient time.
- (8/24) Welcome to the Math 215a webpage! This course will be similar to
what I taught two years ago.
However we will do a bit more with smooth manifolds, and so now Bredon
is a required text in addition to Hatcher. Also, unlike last time,
homework will be graded by a GSI instead of by fellow students; since
you will not have to spend time grading each other's homework
assignments, I will give you a few more exercises to do.
Instructor: Michael Hutchings, 923 Evans, [my last name with
the last letter deleted]@math.berkeley.edu. Tentative office
hours (may be rescheduled some weeks): Wednesday 9-12. Outside of
office hours, the best way to reach me is to send me an email. I
generally check email once per day.
GSI: Qin Li. Office hours: Friday 3-5pm, room 935 Evans.
Lectures: MWF 1-2, room 3 Evans.
Prerequisites: The only formal requirements are some basic
algebra, point-set topology, and "mathematical maturity". However,
the more familiarity you have with algebra and topology, the easier
this course will be. I think that all the point-set topology we will
need (and a lot more) is reviewed in Bredon, Chapter I, Sections 1-13.
Algebraic topology seeks to capture key information about a topological
space in terms of various algebraic and combinatorial objects. We
will construct three such gadgets: the fundamental group, homology
groups, and the cohomology ring. We will apply these to prove various
classical results such as the classification of surfaces,
the Brouwer fixed point theorem, the Jordan curve theorem, the
Lefschetz fixed point theorem, and more.
An important topic related to algebraic topology is differential
topology, i.e. the study of smooth manifolds. In fact, I don't think
it really makes sense to study one without the other. So without
making differential topology a prerequisite, I will emphasize the
topology of manifolds, in order to provide more intuition and
Homework is essential for learning the material, and will form
the basis for your grade. Assignments will be given every 1.5-2 weeks
and will be posted here.
Allen Hatcher, Algebraic topology. This is a great book. You
can download it for free from this page,
which also has some additional material. However no single book can
do everything that one might want, and this book does very little with
smooth manifolds. So we will also be using:
Glen Bredon, Topology and Geometry. This book does a lot more
with smooth manifolds than Hatcher, and I also prefer its treatment
of some of the topics that are covered in Hatcher. If you don't
want to buy it, it is on reserve in the library.
Tentative course outline (There is a lot of material here, so
we might not get to all of it, and I will refer to the textbooks for
details of some of the proofs. My goal will be to explain the
- Introductory lecture: Questions which algebraic topology can answer (or try to).
- Classification questions (manifolds, knots...)
- Topological existence results (e.g. fixed point theorems)
- The fundamental group and covering spaces. (mainly from
Hatcher, Chapter 1; also Bredon, Chapter III)
- The fundamental group.
- Definition and basic properties.
- Basic examples: R^n, S^1 (by lifting to R),
products, S^n for n>1 (by using
smooth approximation to miss a point, or by cutting S^n into two
simply connected pieces whose intersection is connected).
- Applications of the fundamental group of the circle:
- The Brouwer fixed point theorem and Borsuk-Ulam theorem in
- The so-called fundamental theorem of algebra.
- The Seifert-van Kampen theorem for the fundamental group of
the union of two spaces.
- The fundamental group of a CW complex with one 0-cell has a
presentation with one generator for each 1-cell and one relation
for each 2-cell.
- The braid group. (Not in either
of our textbooks.)
- The trefoil knot is knotted and the Hopf link and Borromean
rings are linked. (Not discussed at all in Hatcher, and not much
in Bredon either.)
- Covering spaces.
- Definition and discussion of basic examples, including the
finite covers of S^1 and permutations, coverings of the torus, the
covering of any genus g>1 surface by the hyperbolic plane, the
canonical double covering of any manifold by an oriented manifold,
and the non-example of branched covers of Riemann surfaces.
- Lifting paths to a covering space; the lifting criterion for maps
to a covering space.
- The classification of covering spaces over a reasonable space.
- An application to algebra: any subgroup of a free group is free.
- Brief introduction to higher homotopy groups: definition, easy
properties, introduction to pi_n(S^n) (which will be discussed more
later), why homotopy
groups are hard in general.
- Homology. (Hatcher, Chapter 2; Bredon, Chapter IV)
- Introducing singular homology.
- Warmup definition: simplicial homology of a Delta-complex.
- Main definition: singular homology of a topological space.
- H_0 is a direct sum of Z's, one for each path component.
- Computation of the homology of a contractible space, using
cones over simplices.
- H_1 of a path connected space is the abelianization of pi_1.
- Homology is a homotopy invariant.
- Mayer-Vietoris sequence:
- statement and simple examples, including the homology of S^n.
- A bit of homological algebra: a short exact sequence of chain
complexes induces a long exact sequence in homology.
- Proof of Mayer-Vietoris sequence using subdivision lemma.
- Understanding the connecting homomorphism in the
- Application: proof of the generalized Jordan curve theorem.
- An interesting example: linking numbers.
- How to compute the homology of a CW complex.
- Relative homology; long exact sequences of a pair and a triple;
excision; isomorphism between relative homology and reduced homology
of the quotient
for "good pairs".
- Equivalence of simplicial and singular homology.
- Extensive discussion of the degree of a map from a sphere to itself.
- Cellular homology of a CW complex.
- Application to the "hairy ball theorem".
- Isomorphism between singular and cellular homology of a CW
- Examples including real and complex projective space.
- Euler characteristic, and Lefschetz fixed point theorem for
- Homology with coefficients, Tor, and the universal coefficient
- The Kuenneth formula for the homology of the product of two
spaces, via the Eilenberg-Zilber theorem.
- Cohomology. (Hatcher, Chapter 3; Bredon, Chapters V and VI)
- Definition of cohomology, and universal coefficient theorem for
- Definition of the cup product.
- Statement of Poincare duality.
- Computation of the cohomology ring of projective spaces.
- Application: the Borsuk-Ulam theorem and the Ham Sandwich Theorem.
- Important ingredients in the proof of Poincare duality, including the
orientation bundle, compactly supported cohomology, direct limits,
and the cap
- Review/crash course on smooth manifolds.
- If time permits, review/crash course on de Rham cohomology, and
isomorphism of the latter with singular cohomology with real coefficients.
- Statement of the geometric interpretation of cup product via
intersection theory in a smooth manifold.
- Application: the Lefschetz fixed point theorem on a smooth
actually counts fixed points of a generic smooth map with signs,
rather than merely proving
that a fixed point exists).
- Poincare-Lefschetz duality for a manifold with boundary.
- Application: the boundary of a 4n+1 dimensional oriented
compact manifold has signature zero.
- Alexander duality.
- Application: the Klein bottle does not embed in R^3 (and more
generally, a nonorientable closed n-manifold does not embed in
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