Assignments
Assignment 1:
Due Wednesday, January 21, 1998 in class.
Write out the solution to problem 2, page
7 of Munkres.
A solution is here.
Assignment 2:
Due Friday, January 23, 1998 in class.
Show that the collection of finite simplicial
complexes is countable.
More precisely, show that there is a countable list
of finite simplicial complexes such that any finite
simplicial complex is simplicially isomorphic to one
and only one complex on your list.
A solution is here.
Assignment 3:
Due Monday, February 9, 1998 in class.
- Compute the homology of the 1-skeleton of the 3-simplex.
- Using that the homology of an r-simplex is 0 in dimensions
greater than 0 (and Z in dimension 0), compute the homology
of the boundary of it.
Assignment 4:
Problem 2 on page 175 of the text.
Assignment 5:
Problem 3 on page 213 of the text.
A solution is here.
Assignment 6:
Problems 1 and 3 on page 221 of the text.
A solution is here.
Midterm:
A copy of the midterm is here.
It is due in class on Monday, April 6.
Assignemnt 7:
For Monday, April 27, prove part b of the Acyclic Models theorem.
Let $G$ and $G^\prime$ be functors from a category $C$ to
the category of augmented chain complexes. Let
$M$ be a set of models. Assume $G^\prime$ is acyclic
for $M$ and $G$ is free.
If $T$ and $T^\prime$ are natural transformations from
$G$ to $G^\prime$, show that there exists a natural
chain homotopy from $T$ to $T^\prime$.
Even better, show that the natural chain homotopy exists
under the weaker assumption that for all $\alpha\in J_p$,
$H_p(G^\prime(M_\alpha))=0$ rather than acyclicity. If
$p=0$ use $\epsilon H_0(G^\prime(M_\alpha))\to Z$ is an
isomorphism.