## 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.