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\begin{document}
\pagestyle{fancy} \lhead{1301 Topology II, Assignment 4} \rhead{Due
  date: March 22, 2013} \cfoot{}
\noindent {\bf Reading:} Hatcher \S 2.2.

\begin{exercise}
Hatcher, Section 2.2 Exercises 8, 20, 21, 22.
\end{exercise}

\begin{exercise}
Let $E$ be a double complex, meaning that $E$ is a $\ZZ\times \ZZ$ graded vector space,
\[
E = \bigoplus_{(i,j)\in\ZZ\times\ZZ} E^{i,j},
\]
equipped with two operators $\del,\delbar$ of degrees $(1,0)$ and $(0,1)$ respectively, such that $\del^2 = \delbar^2 = 0$ and $\del\delbar = \delbar\del$.

\vspace{5cm}

\begin{enumerate}
\item Produce from $E$ another $\ZZ\times\ZZ$ graded vector space $E_1$, by taking $\delbar$ cohomology:
\[
E_1^{i,j} = \frac{\ker \delbar|_{E^{i,j}}}{\delbar(E^{i,j-1})}.
\]
Prove that both $\del, \delbar$ induce maps $\del_1,\delbar_1$ of degree $(1,0), (0,1)$ respectively on $E_1$, that $\delbar_1=0$, and that $\del_1^2 = 0$.

\item Produce from $E_1$ another $\ZZ\times\ZZ$ graded vector space $E_2$, by taking $\del_1$ cohomology:
\[
E_2^{i,j} = \frac{\ker \del_1|_{E_1^{i,j}}}{\del_1(E_1^{i-1,j})}.
\]
As before, the map induced by $\del_1$ is zero, but prove that the following procedure defines a well-defined map $\del_2:E_2\to E_2$ of degree $(2,-1)$:

\center{\it Let $a\in E^{i,j}$ be such that $\delbar a = 0$ and $\del a = \delbar b$.  Then $\del b \in E^{i+2, j-1}$.
}

\vspace{4cm}


\flushleft Finish by proving that $\del_2^2 = 0$.
\end{enumerate}

\end{exercise}


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