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\title{Homework 2: Due Tuesday 2/5/19\\
MTH 961 W19}
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\begin{enumerate}
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% Question 1
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\item Give a CW decomposition of $\bC P^{\infty}$. Compute the cellular cohomology $H^*(\bC P^{n},R)$ with coefficients in a ring $R$, as a graded ring, for $1\le n\le \infty$. Also, compute the cellular cohomology of $H^*(\bR P^{n},\bZ /2)$ as a graded ring for $1\le n\le \infty$. Prove that there is no map $\bR P^n\to \bR P^m$ with $n>m$ that induces an isomorphism on $\pi_1$.
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% Question 2
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\item For each element $[f]\in \pi_3S^2$ compute the graded cohomology ring of the cofiber of $f$, $Cf$, with coefficients in $\mathbb{Z}$.
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% Question 3
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\item List all homotopy types of CW complexes with one $0$-cell, one $2$-cell, and one $3$-cell. Do the same for all CW complexes with one $0$-cell, one $2$-cell, and one $4$-cell.
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% Question 4
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\item Let $n\ge 1$ and let $\pi$ be an abelian group. Construct a CW complex $M(\pi,n)$ such that $\tilde{H}_n(X,\bZ)=\pi$ and $\tilde{H}_q(X,\bZ)\cong 0 $ for $q\ne n$. (Hint: construct $M(\pi,n)$ as the cofiber of a map between wedges of spheres. The spaces $M(\pi,n)$ are called Moore spaces.)
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\item Let $n\ge 1$ and let $\pi$ be an Abelian group. Construct a connected CW complex $K(\pi,n)$ such that $\pi_n(X)=\pi$ and $\pi_q(X)=0$ for $q\ne n$.
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% Question 6
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\item Given abelian groups $G$ and $H$ and CW complexes $K(G,n)$ and $K(H,n)$, show that the map $H^n(K(G,n);H)\cong [K(G,n),K(H,n)]\to \text{Hom}_{\mathbb{Z}}(G,H)$ given by sending $f$ to $f_*:\pi_nK(G,n)\to \pi_nK(H,n)$ is a bijection.
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