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  \newtheorem*{rrt}{Rank Rigidity Theorem}{}

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  \begin{document}

  \centerline{\Large {Cohomology}}

\begin{enumerate}[1.]

\item 
Let $p:\tilde X\to X$ be a double cover. Show that there is a long
exact sequence (the Gysin sequence) of the form
$$\cdots \to H^k(X;\Z/2\Z)\to H^k(\tilde X;\Z/2\Z)\to H^k(X;\Z/2\Z)\to
H^{k+1}(X;\Z/2\Z)\to\cdots$$
From this deduce that $H^i(\R P^n;\Z/2\Z)\cong \Z/2\Z$ for
$i=0,1,\cdots,n$ and is 0 otherwise.

Hint: $0\to C_n(X;\Z/2\Z)\to C_n(\tilde X;\Z/2\Z)\to C_n(X;\Z/2\Z)\to
0$.

\item Let $X$ be a space and suppose that $H_0(X;\Z)\cong\Z$,
  $H_1(X;\Z)\cong H_2(X;\Z)\cong\Z\oplus\Z/6\Z$, $H_3(X;\Z)\cong
  \Z$ and $H_i(X;\Z)=0$ for $i>3$. Compute $H_i(X;\Z/2\Z)$,
  $H^i(X;\Z)$, $H^i(X;\Z/2\Z)$ for all $i$.

\item Compute the ring structure on $H^*(X;\Z/n\Z)$ where $X$ is the Moore
  space $X=S^1\cup_f e^2$ with the attaching map $f$ of degree $n$.
  Hint: First compute cohomology groups $H^i(X;\Z/n\Z)\cong\Z/n\Z$ for
  $i=1,2$ using the cellular chain
  complex. Then for the cup product use a $\Delta$-complex structure
  on $X$ and the definition of the cup product. Instead of doing a lot
  of linear algebra find a 1-cycle and a 1-cocycle whose Kronecker
  pairing over $\Z/n\Z$ is $1\in\Z/n\Z$. This gives a generator in
  $H^1(X;\Z/n\Z)$. Note that the case
  $n=2$ recovers $H^*(\R P^2;\Z/2\Z)$.

\item Let $X,Y$ be connected $\Delta$-complexes and consider $Z=X\vee Y$,
  the wedge sum along vertices in $X$ and $Y$. Show, by using
  functoriality properties and the definition of the cup product (just the
  easy part that if $\phi,\psi$ are cocycles that don't evaluate 0 on
  the faces of the same simplex then $\phi\cup\psi=0$), that the cohomology ring $H^*(Z;\Z)$ is
  obtained by taking the direct sum $H^*(X;\Z)\oplus H^*(Y;\Z)$ of the
  two rings (with the product of elements in distinct coordinates 0)
  and then identifying the units $1_X=1_Y$. Hint: Consider inclusions
  $X,Y\hookrightarrow Z$ and retractions $Z\to X,Y$.

\item Consider the map $S_2\to T_1\vee T_2$ from the genus 2 surface
  to the wedge of two tori obtained by collapsing a separating
  curve. Use your knowledge of the ring structure on the cohomology of
  the torus, Exercise \#4, and cellular chain complexes to
  deduce the cohomology ring of $S_2$. Cf. Exercise \#1 in Hatcher
  Section 3.2.

In addition, do \#6,7,10,11,18 from Hatcher Section 3.2.

 \end{enumerate}

\end{document}
