Math 215b: Algebraic topology
UC Berkeley, Spring 2004
[My last name with the last letter deleted]@math.berkeley.edu
Office phone: 510-642-4329.
Office: 923 Evans.
Office hours: by appointment for now; I
may pick a regular time a little later.
As of 2/12, we are moving to 41 Evans. (The old room was 65 Evans.)
The time is TTh, 11:00 - 12:30.
About the course
In 215a we studied the fundamental
group, homology, and cohomology, from Chapters 1-3 of Hatcher's
book "Algebraic Topology". (We skipped over some of what Hatcher
did, and we did a few things which Hatcher didn't, particularly in
connection with differential topology. Bredon's book "Topology and
Geometry" is a good reference for those additional topics.)
With the basics of algebraic topology established, in 215b we will
study a selection of more advanced topics, which nonetheless are in my
opinion very important and useful in geometry and topology. I will
emphasize applications to the geometry of smooth manifolds. We will
do as much of the following as time permits (not in exactly this
- Higher homotopy groups and obstruction theory.
- Bundles and
- Basic Morse theory and applications to
- Spectral sequences.
- As time
permits, possible additional topics to be determined, such as:
- Secondary invariants such as Reidemeister torsion, Alexander
polynomial, Massey products, etc.
- Equivariant cohomology.
You should already have a basic graduate level algebraic topology text
such as Hatcher and/or Bredon; this will help with the first part of
the course. The beautiful book "Characteristic classes" by Milnor and
Stasheff is strongly recommended reading for a substantial part of the
course. "Differential forms in algebraic topology" by Bott and Tu
gives a very nice exposition relating to some of the topics in the
course. I will give more references for specific topics we cover as
the course progresses.
I will suggest some homework exercises; these will not be graded,
although I am happy to discuss how to solve them.
The only course requirement will be to write an expository paper on
some specific topic of interest in algebraic topology. This should be
about 10 pages or less. Inspired by Prof. Weinstein, I will ask
students to anonymously referee each other's papers and then revise
them. Click here for more details.
Summaries of the lectures with references as appropriate will be
- Defined higher homotopy groups. (These are discussed in most
algebraic topology textbooks.)
- Sketched a geometric proof of the
Hurewicz isomorphism theorem. (I don't know a reference for this
proof unfortunately. Later we will see a much shorter proof using the
Leray-Serre spectral sequence. I think that the latter proof actually
has the same underlying geometric content as the former.)
Started discussing the dependence of pi_k on the choice of basepoint,
by showing that a path between two base points induces an isomorphism
between the corresponding pi_k's, which depends only on the homotopy
class of the path etc. (If you want to be fancy, pi_k is a functor on
the fundamental groupoid of X.)
Introduced the mapping torus of a homeomorphism, an example where the
action of pi_1 on pi_k can be nontrivial.
- Introduced fiber
bundles. Clutching construction, Hopf fibration. Started on homotopy
properties of fiber bundles. (cf. Hatcher section 4.2. As I recall,
Steenrod's book The topology of fiber bundles is beautiful and
might be a good reference for what we are discussing.)
- Continued with homotopy properties, defining the pullback of a
fiber bundle and proving that any fiber bundle over a contractible CW
complex is trivial.
- Introduced the long exact sequence in homotopy groups for a fiber
bundle. Did basic examples. Discussed the Hopf invariant.
- Defined homotopy lifting property and Serre fibrations, showed
that every fiber bundle is a Serre fibration. Used this to define the
connecting homomorphism in the long exact sequence.
- Proved exactness of the long exact sequence in homotopy groups
associated to a Serre fibration. One more example: the path fibration
of a topological space.
- Proved that for any CW complex X, there
is a canonical isomorphism [X,S^1]=H^1(X;Z). This is a prototypical
obstruction theory argument; we will do several similar proofs later.
- Class was cancelled on Tuesday Feb. 3, because I had jury duty.
Fortunately they called in more people than they needed and I was not
selected. We will have a makeup
lecture later at a time to be determined.
- Defined section of a fiber bundle and orientation of a sphere
- For a sphere bundle E over a CW complex B, defined
w_1(E) in H^1(B;Z/2), the obstruction to orienting E.
- Started to
define the Euler class of an oriented sphere bundle, which is the
primary (but sometimes not the only) obstruction to finding a section.
- Discussed the Euler class in detail from the viewpoint of
- Proved that an oriented S^k bundle over a CW
complex has a section over the k+1 skeleton iff its Euler class
- Showed that the Euler class gives an isomorphism
between the set of oriented circle bundles over a given CW complex B
(modulo orientation-preserving bundle isomorphism) and H^2(B;Z).
Made some brief remarks on what a characteristic class is, what the
primary obstruction to finding a section of a general fiber bundle
looks like, and Eilenberg-Maclane spaces.
- Whitehead's theorem: if a map between CW complexes induces an
isomorphism on all homotopy groups, then it is a homotopy equivalence.
- Basic stuff about vector bundles. (See the first few chapters of
Milnor and Stasheff.)
- Introduction to Stieffel-Whitney classes; see Milnor-Stasheff
- A bit about spin structures and the obstruction to their existence.
- Discussed how Stieffel-Whitney numbers give obstructions to
unoriented cobordisms. (See M-S ch. 4.)
- Vector bundles modulo
isomorphism over (a reasonable space) B are the same as homotopy
classes of maps from B to the classifying space (the infinite
Grassmannian). Therefore the ring of all
characteristic classes is the cohomology ring of the classifying
space. (See M-S ch. 5.)
- Discussed the cell decomposition of the real and complex
Grassmannians in terms of Schubert varieties. (See M-S ch. 6.)
- As a fun example, explained (in terms of intersection theory in
the Grassmannian G_2(C^4)) why given four generic lines in CP^3
there are two lines meeting all four of them.
- We showed that the
Z/2 cohomology of the real infinite Grassmannian is the polynomial
ring over the Stiefel-Whitney classes, assuming that the
Stiefel-Whitney classes exist. (See M-S chapter 7.)
- Discussed one definition of Stiefel-Whitney classes in terms of
obstruction theory. (See M-S chapter 12.) However it is hard to
prove the Whitney sum formula from this definition.
- Introduced the Chern classes of a complex vector bundle, their
basic properties, and their interpetation in terms of obstructions
when the base is a CW complex. (See
M-S ch 14.)
- Discussed how in the smooth case, the Euler class of an oriented
real vector bundle is Poincare dual to the zero set of a generic section.
(I don't know a reference for this, but I think it is important
to understand the basic intuition here. Maybe I'll try to write
- Proved the Thom isomorphism theorem for an oriented real vector
bundle over any base. (See M-S chapter 10 for the details that I
- Showed that (in the category of smooth compact
oriented manifolds without boundary) the Thom class of the normal
bundle (which is diffeomorphic to a tubular neighborhood) of a
submanifold is Poincare dual to the fundamental class of the
- Used this to finally explain why given two
transverse submanifolds, the Poincare dual of their intersection is
the cup product of their Poincare duals. (This is in Bredon.)
- Used the Thom isomorphism to redefine the Euler class (over any
base) and give simple proofs of its basic properties. (See M-S
- Introduced the Gysin sequence and used it to show
that the above definition of the Euler class agrees with the
obstruction-theoretic definition over a CW complex. (See M-S ch. 12.)
- Showed how the Gysin sequence can be used to prove uniqueness of
the Chern classes and give a definition of them from which all of the
axioms can be proved. (See M-S ch. 14 for more about this.)
Next week we will put characteristic classes aside and start on
spectral sequences! If you want to learn more about characteristic
classes, the rest of Milnor and Stasheff is good reading. For those
of you with an interest in geometry, I especially recommend Appendix
C, explaining the beautiful relation between characteristic classes
- More properties of Chern classes, the Chern character, and the
Splitting Principle. (The proof of the latter that I
presented is from Bott and Tu.)
- Brief remarks about the Pontrjagin classes.
Started discussing spectral sequences, which I will explain by
successive approximations; hopefully it will eventually start to make
- Today we discussed how to compute the homology of a
filtered chain complex by successive approximations. (See
e.g. Griffiths and Harris pp. 438-442.)
- Clarified the basics of spectral seuqences from last time.
- Introduced the Leray-Serre spectral sequence and did some simple
computations using it.
- Re-proved the Hurewicz isomorphism theorem using the Leray-Serre
spectral sequence. (For this and many more applications of the
Leray-Serre spectral sequence, see Bott and Tu.)
Discussed the relation between the Euler class of a sphere bundle and
the Leray-Serre spectral sequence.
Explained homology with twisted coefficients, discussed the
Leray-Serre spectral sequence over a non-simply-connected base, and
considered the example of a mapping torus.
- If a filtered cochain complex has a product respecting the
filtration such that the differential is a derivation, then the
differentials in the associated spectral sequence are derivations with
respect to induced products. In particular, the differentials in the
Leray-Serre spectral sequence are derivations with respect to the
(twisted) cup product.
- For an oriented sphere bundle, the
differential in the spectral sequence is cup product with the Euler
- We proved the Leray-Hirsch theorem.
- Showed how the Leray-Serre spectral sequence can give some
information about homotopy groups of spheres. For more on this topic
see Bott-Tu or Spanier.
- Discussed the Mayer-Vietoris spectral
sequence and its uses.
- Explained the "universal coefficient spectral sequence". As an
example, used it to give a somewhat more difficult than usual
computation of the homology of a two-torus or a wedge of
two circles. I don't know a good reference for this but it is a
special case of more general homological algebra.
- *** MASSIVE SHIFT OF GEARS HERE ***
- Introduced Morse theory.
- (4/13) Discussed how classical Morse theory gives rise to handle
decompositions of manifolds. For more details see:
- Milnor's book "Morse theory", chapters 1,2,3.
and Kirby calculus" by Gompf and Stipsicz, early parts of chapters 4,
5, and 9 and the references therein. (The other parts of this book
are full of fun stuff, too.)
- Some parts of my lecture notes on modern Morse theory might
also be useful (modulo the mistakes therein).
(Isn't this stuff fun? Maybe I should have started this earlier in
the course. I think all these explicit low-dimensional pictures really bring
algebraic topology to life.)
- Used handle calculus to classify compact oriented surfaces.
- Going up a dimension, discussed Heegard splittings of
- Going up another dimension, introduced Kirby calculus for
3-manifolds (and 4-manifolds that they bound).
- Explained how to compute the homology of a manifold in terms of a
handle decomposition. This is closely related to (and with more
technical work can be realized as a special case of) the cellular
homology of a CW complex.
- Started to sketch the proof that if a
closed smooth n-manifold with n at least 6 is simply connected and has
the homology of S^n, then it is homeomorphic (although not necessarily
diffeomorphic) to S^n. The argument is essentially the h-cobordism
theorem; see Gompf-Stipsicz chapter 9 for an outline, and Milnor's
book "Lectures on the h-cobordism theorem" for full details.
- Explained Whitney's cancellation lemma for getting rid of
intersection points of submanifolds in sufficiently high dimension.
(Cf. Milnor's Lectures on the h-cobordism theorem.)
- Mostly finished the explanation of the proof of the
higher-dimensional Poincare conjecture.
- Discussed Dehn surgery, examples thereof, and why every oriented
3-manifold can be obtained from Dehn surgery on a framed link in S^3.
- Introduced Morse homology, which we will explain in more detail
- Explained the definition of Morse homology and the signs in the
differential, and did simple examples.
- Proved that the differential is well-defined and has square zero,
modulo the technical theorem about the compactification of the moduli
space using broken flow lines.
Explained the a priori proof that Morse homology is a topological
invariant, in terms of continuation maps and chain homotopies.
Showed how Poincare duality is very easy in Morse homology.
Explained the isomorphism between Morse homology and singular homology.
- (5/11) Discussed classical applications of Morse theory relating
geodesics to the topology of the loop space, including:
(For details see Milnor's book "Morse theory".) There are lots of
other things I would have liked to do in this course but didn't have
time for. Some of these are covered in the term papers, which I will
post here (given authorial permission).
- the Freudenthal suspension theorem
- existence theorems for geodesics
- Bott periodicity for the unitary groups
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