## 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.