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16
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Does the following condition imply the homotopy type of a wedge of spheres?
You can take $S^1\vee S^2\vee S^3$ and then use the Hopf fibration from $S^3$ to $S^2$ as the attaching map for a 4-cell onto the 2-cell. This has to have the cohomology (and homotopy) as you describ …