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\begin{document}

\title{MTH 628 Homework 1}

\author{Witold  Hurewicz}

\date{February 14, 2017}

\maketitle


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% PROBLEM 1
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section*{Problem 1}

In order to solve this problem it will be convenient to start with the following fact:

\begin{lemma}
\label{HELPFUL LEMMA}
For any space $\widetilde H_{q}(X)\cong \widetilde H_{q-1}(\Sigma X)$
\end{lemma}


\begin{proof}
Here is the argument...
\end{proof}

Using Lemma \ref{HELPFUL LEMMA} we will prove that the following is true:

\begin{theorem}
\label{BIG THEOREM}
The following diagram commutes:

$$
\begin{tikzpicture}
% FIRST DESCRIBE VERTICES IN THE DIAGRAM
\matrix (m) 
[matrix of math nodes, row sep=2em, column sep=3em, text height=1.5ex, text depth=0.25ex]
{
\vdots & \vdots & \vdots \\
A_{n} & B_{n} & C_{n}  \\
A_{n-1} & B_{n-1} & C_{n-1}  \\
\vdots & \vdots & \vdots \\
};
% NEXT DESCRIBE ARROWS BETWEEN VERTICES
\path[->, thick, font=\scriptsize]
(m-1-1) edge node[anchor=east]  {$\partial$} (m-2-1)
(m-1-2) edge node[anchor=east]  {$\partial$} (m-2-2)
(m-1-3) edge node[anchor=east]  {$\partial$} (m-2-3)
(m-2-1)
edge node[anchor=south] {$f_{n}$} (m-2-2)
edge node[anchor=east]  {$\partial$} (m-3-1)
(m-2-2)
edge node[anchor=south] {$g_{n}$} (m-2-3)
edge node[anchor=east]  {$\partial$} (m-3-2)
(m-2-3)
edge node[anchor=east]  {$\partial$} (m-3-3)
(m-3-1)
edge node[anchor=north] {$f_{n-1}$} (m-3-2)
edge node[anchor=east] {$\partial$} (m-4-1)
(m-3-2)
edge node[anchor=north] {$g_{n-1}$} (m-3-3)
edge node[anchor=east] {$\partial$} (m-4-2)
(m-3-3)
edge node[anchor=east] {$\partial$} (m-4-3)
; 
\end{tikzpicture}
$$ 


\end{theorem}


\begin{proof}[Proof of Theorem \ref{BIG THEOREM}]
We argue by induction with respect to $n$. For $n=0$ ...
\end{proof}

Exercise 2.1 follows immediately from Theorem \ref{BIG THEOREM}. Indeed, assume that... 


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% PROBLEM 2
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\section*{Problem 2}

We will prove this by contradiction. Assume that....





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