The space $ A$ is called homotopy dominated by the space $ X$ if there are maps $ f:A\longrightarrow X$ and $ g:X\longrightarrow A$ so that $ g\circ f\simeq id_A$ .
Question: If $ A$ is homotopy dominated by $ X_1\vee X_2$ , then is $ A$ of the form $ A_1 \vee A_2$ (up to homotopy equivalent) where $ A_i$ is homotopy dominated by $ X_i$ for $ i=1,2$ ?
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