dim 8
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Let $\varphi: S^7\to S^4$ be the Hopf map. Let $n\geq 1$ be an integer, and the map $c_n:S^7\to S^7\vee\cdots\vee S^7$ divides the sphere $S^7$ into $n$ spheres. Define a map
$$f_n:S^7\xrightarrow{c_n}S^7\vee\cdots\vee S^7\xrightarrow{\varphi\vee\cdots\vee\varphi}S^4.$$
Prove that the space $X_n = S^4\cup_{f_n}D^8$ is homotopy equivalent to a closed compact manifold of dimension $8$ if and only if $n = 1$.
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