Is it true that if $A_1\times A_2\times ... \times A_n = B_1\times B_2\times .. \times B_m$, where $A_i, B_j$ are homotopy types of connected complexes not decomposable into a product, then the multisets $A_i$ and $B_j$ coincide? What if it is limited only to finite complexes?
Is it true that if $A_1\wedge A_2 \wedge .. \wedge A_n = B_1\wedge B_2 \wedge .. \wedge B_m$, where $A_i, B_j$ are homotopy types of connected pointed complexes not decomposable into a smash product, then the multisets $A_i$ and $B_j$ coincide? What if it is limited only to finite complexes?
It is clear that finite complexes decompose into a finite number of indecomposable (with smash product, the connectedness increases, and with a Cartesian product, the homology groups increase).
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