Math 636, Algebraic Topology
Here are some review problems to prepare for the final
exam, at 10:30 on Monday June 7. We will meet to discuss them on Sunday at 2.
Homework due Monday 4/5:
- Compute the homology of RP2 x RP2. Does the answer
surprise you?
- From Hatcher's book, page 184, problems 2 and 4.
- Prove or disprove: any map from S2 v S4 to itself
must have a fixed point.
We will start the quarter with cohomology. Read section 3.1 in Hatcher's book
to prepare.
Dan Dugger will be teaching during the second week and will cover universal coefficient
theorems from an algebraic point of view.
Homework due Monday, 4/19:
- Compute the bi-complex associated to and resulting homology of
the product of M(Z/3, 1) x RP3.
- Hatcher, page 204: 3, 5, 7.
Week three, 4/12-4/16:
On Monday and Wednesday we will develop the Kunneth theorem
from the more computational, geometric,
and "philosophical" points of view. On Friday we will start in earnest
developing the cup product structure on cohomology.
Week four, 4/19-4/23: We will introduce the cup product and then go through
the cohomology of the two-holed torus explicitly to compute cup products.
We will be thorough, developing it in such a way so as to see how
basic definitions work and to see concepts such as perfect pairings which
will be helpful when we cover Poincare duality.
There will be a take-home midterm handed out on Wednesday 4/28, due on
Monday 5/3.
During weeks five and six we work on properties of the cup
product, including its commutativity.
We will take our time in stating Poincare duality in its
full generality and proving it carefully; there is a lot of
good mathematics just in the ingredients.
Week seven: We will finish the verification of Poincare
duality, as phrased in terms of cap products, in the case of the
torus as a delta complex.
We will then move quickly through
the naturality of the cap product and then on to talking about
orientations in terms of (local) homology. On (the latter part of)
Wednesday and (all of) Friday the first group, John C., Chad G., Stephen J.,
and Max A., will present Theorem 3.26, which is rightfully viewed as the first
case of Poincare duality where the homology group in question lies
in the dimension of M.
Week eight: I will start the week discussing cohomology with compact supports
on Monday. On Wednesday I will set up the cap product needed for cohomology
with compact supports, state the version of Poincare duality we are ultimately
proving, namely Theorem 3.35, and then hand the stage over to the second
group, Bill K., Dawn A., Lene C., Tim S. and Sammy B.,
who will show how the proof follows readily once we have Lemma 3.36.
It is also likely that we will have a homework assignment. Stay tuned.
Week nine: Finally, on Monday and Wednesday of this week, the third group,
Seth A., Michael R., Jonathan B., Nathan C., Aaron L.,
will present the proof of the all-important, and technical, Lemma 3.36.
From there, through the rest of the term for the most part, I will give consequences of
this general duality theorem, including Alexander duality and computations
of cohomology rings of manifolds.