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\title{Math 490 Homework 1\\ Due: February 2}

\begin{document}

\maketitle

You're strongly encouraged to write your homework in \LaTeX. I won't require it for the first assignment, but I likely will for the rest of the assignments.

\begin{enumerate}
\item Let $M$ be an abelian group. Show that there is a unique way to give $M$ the structure of a $\bZ$-module ($\bZ$ is the ring of integers).


\item Let ${\bf M}$ be a complex of finite-dimensional vector spaces:
\[
 M_n \to M_{n-1} \to \cdots \to M_1 \to M_0
\]
Show that
\[
\sum_{i=0}^n (-1)^i \dim M_i = \sum_{i=0}^n (-1)^i \dim {\rm H}_i({\bf M}).
\]

\item
    \begin{enumerate}
    \item Let $G$ be the following oriented graph (I've labeled the vertices with coordinates).
      \begin{align*}
\xymatrix{ (0,1) \ar[r]^-{\alpha_1} \ar[d]^-{\alpha_3} & (1,1) \ar[r]^-{\alpha_2} \ar[d]^-{\alpha_4} & (2,1) \ar[d]^-{\alpha_5} \\ (0,0) \ar[r]^-{\alpha_6} & (1,0) \ar[r]^-{\alpha_7} & (2,0) }
      \end{align*}
Write down the matrix representing the boundary homomorphism for $G$ and compute a basis for ${\rm H}_1(G)$.


\item More generally, pick positive integers $m,n$ and let $G_{m,n}$ be the graph whose vertices are points $(i,j)$ with $0 \le i \le m$ and $0 \le j \le n$. (The previous part concerns $G_{2,1}$.) All of the edges are between vertices ``next to each other'', i.e., $(i,j)$ is connected to both $(i+1,j)$ and $(i,j+1)$. 

Pick whatever orientation you like for the edges (explain it) and then compute a basis for ${\rm H}_1(G_{m,n})$. Also, give a simple formula for the size of the basis.

Part of the problem is to figure out a good way to label the edges (the way I did it in part (a) isn't so good -- they don't have to all be called $\alpha$!).
    \end{enumerate}
\end{enumerate}

\newpage

Hints:

\begin{enumerate}
\item You know that you have to define $1 \cdot m = m$. What would $2 \cdot m$ have to be? $-1 \cdot m$?
\item Try the case $n=1$ first and then $n=2$. The review exercises will help here.
\item This problem can be solved using the results we proved in class, and is more about coming up with a good way to develop your own notation. 

For the general formula: if you can't deduce it, try fixing $m$ and varying $n$ and computing examples and see if you can find a pattern.
\end{enumerate}

\end{document}