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\title{Homework 3: Due Tuesday 2/19/19\\
MTH 961 W19}
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\item Prove that all the maps in the derived exact couple of an exact couple are well-defined and that the derived exact couple of an exact couple is again an exact couple.
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\item Compute $H_*(\Omega S^n;\mathbb{Z})$ and compute $H^*(\Omega S^3;\mathbb{Z})$ as a graded ring.
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\item Compute $H^*(K(\bZ,n);\bZ)$ and $H^*(K(\bZ/p,n);\bZ/p)$ for $n=1,2,3$ as a ring for $p$ any prime. (Compute as much as you can.)
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\item A \it{two-stage Postnikov system} \rm is a space $X$ which is the fiber of a map between Eilenberg-MacLane spaces. List all homotopy types of two-stage Postnikov systems with one $K(\bZ/2\bZ,1)$ and one $K(\bZ/2\bZ,2)$.
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% Question 5
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\item A \it{three-stage Postnikov system} \rm is a space $X$ which is the fiber of a map from a two-stage Postnikov system to an Eilenberg-MacLane space. List all homotopy types of three-stage Postnikov systems with one $K(\mathbb{Z}/2,1)$ one $K(\mathbb{Z}/2,2)$ and one $K(\mathbb{Z}/2,3)$.
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