\documentclass[11pt,notitlepage]{article}
\usepackage{graphicx, datetime, multicol, qrcode,amsmath,amssymb,../picins,import,xypic,
  mathtools, % for \coloneqq
  stmaryrd, % for \sslash
  amsthm % for \theoremstyle
}
\usepackage[margin=0.75in,foot=0.5in,top=0.5in]{geometry}
% Following http://tex.stackexchange.com/a/847/22475:
\usepackage[setpagesize=false]{hyperref}\hypersetup{colorlinks,
  linkcolor={blue!50!black},
  citecolor={blue!50!black},
  urlcolor=blue
}
\usepackage{bbm} % Needed for \bbe.

\def\sheeturl{{\url{http://drorbn.net/26-1301}}}
\def\myurl{http://www.math.toronto.edu/~drorbn}

\def\blue{\color{blue}}
\def\green{\color{green}}
\def\red{\color{red}}

\def\bbe{\mathbb{e}}
\def\bbH{{\mathbb H}}
\def\bbN{{\mathbb N}}
\def\bbQ{{\mathbb Q}}
\def\bbR{{\mathbb R}}
\def\bbZ{{\mathbb Z}}
\def\calA{{\mathcal A}}
\def\calC{{\mathcal C}}
\def\calF{{\mathcal F}}
\def\calH{{\mathcal H}}
\def\calL{{\mathcal L}}
\def\calR{{\mathcal R}}
\def\calS{{\mathcal S}}
\def\calT{{\mathcal T}}
\def\calV{{\mathcal V}}
\def\calX{{\mathcal X}}
\def\calY{{\mathcal Y}}

\def\act{{\hspace{-1pt}\sslash\hspace{-0.75pt}}}
\def\arXiv#1{{\href{http://arxiv.org/abs/#1}{{\tiny arXiv:}\linebreak[0]{\small #1}}}}
\def\eps{\epsilon}
\def\Hom{\operatorname{Hom}}
\def\qed{{\linebreak[1]\null\hfill\text{$\Box$}}}
\def\yellowm#1{{\setlength{\fboxsep}{0pt}\colorbox{yellow}{$#1$}}}
\def\yellowt#1{{\setlength{\fboxsep}{0pt}\colorbox{yellow}{#1}}}

\pagestyle{empty}

\begin{document}
\parindent 0in

\parbox[b]{2in}{\tiny
  \href{http://www.math.toronto.edu/~drorbn}{Dror Bar-Natan}: \href{http://www.math.toronto.edu/~drorbn/classes}{Classes}:
  \newline \href{http://www.math.toronto.edu/~drorbn/classes/\#2526}{2025-26}: \href{http://drorbn.net/26-1301}{MAT 1301 Algebraic Topology}:
}~{\LARGE\bf Homework Assignment 5}
\hfill\parbox[b]{1.5in}{\tiny
  \null\hfill\sheeturl
  \newline\null\hfill Due March 4 at 11:59pm
}~%
\raisebox{0.2em}{
  \qrcode[height=1.2em,level=L,nolink]{drorbn.net/26-1301}
}

\vskip -3mm
\rule{\linewidth}{1pt}
\vskip 5mm

\noindent{\bf Problem 1.} Just in case you thought commutative diagrams cannot get any worse, here's a question to prove you wrong (though actually, it is surprisingly easy).
\begin{enumerate}
\item Define a category $\calS$ of ``short exact sequences of chain complexes''.
\item Define a category $\calL$ of ``long exact sequences''.
\item Construct a functor $\calH\colon\calS\to\calL$. (We've constructed this functor on objects already, so you don't need to do that again. The challenge is to do it on morphisms and to verify that starting from a morphism $\calF$ in $\calS$, its image $\calH(\calF)$ is indeed a morphism in $\calL$).
\end{enumerate}

\vskip 2mm
\noindent{\bf Problem 2.} Given a morphism $f\colon(X,A)\to(Y,B)$, show that the diagram
\[ \xymatrix{
  H_n(X,A) \ar[r]^\delta \ar[d]_{f_\ast} & H_{n-1}(A) \ar[d]_{f_\ast} \\
  H_n(Y,B) \ar[r]^\delta & H_{n-1}(B)
} \]
is commutative.

\vskip 2mm
\noindent{\bf Problem 3.} Given a triple of spaces $B\subset A\subset X$, construct a long exact sequence relating $H_\ast(X,A)$, $H_\ast(A,B)$, and $H_\ast(X,B)$.

\vskip 2mm
\noindent{\bf Problem 4.} Khovanov homology is an invariant of knots obtained by first defining a chain complex $\calC$, and then taking its homology. To show that Khovanov homology does not change when the knot moves, one has to show that pieces of $\calC$ can be cancelled off --- be removed without changing the homology.
\begin{enumerate}
\item Suppose $\calC'$ is an acyclic subcomplex of $\calC$ (``acyclic'' is a different word for ``exact'', or ``having no homology''). Show that $\calC'$ can be cancelled off. Namely, that the quotient $\calC/\calC'$ has the same homology as the original complex $\calC$.
\item Suppose $\calC'$ is a subcomplex of $\calC$, and that the quotient complex $\calC/\calC'$ is acyclic. Then $\calC'$ has the same homology as $\calC$ (roughly, the complement of $\calC'$ can be cancelled off).
\item There is a third theorem along these lines. What does it say?
\end{enumerate}

\end{document}