TITLE: Homology of symmetric product
QUESTION [8 upvotes]: If two distinct topological spaces X and Y have the same homology groups, then their symmetric products also have the same homology?
Do you know if is it true? Do you Know any references for?

REPLY [4 votes]: A stronger statement is true.
If $X$ and $Y$ are spaces such that $H_*(X) \approx H_*(Y)$ (no need to have a map that induces this isomorphism), then $SPX$ and $SPY$ are weakly homotopy equivalent. In particular if $X$ and $Y$ are CW complexes, then $SPX$ and $SPY$ are homotopy equivalent. This can be found in Hatcher: Algebraic topology, (see Hatcher's website).
More precisely: 

By the Dold-Thom theorem $\pi_n(SPX) \approx H_n(X).$ (Thm 4.83 in Hathcher's book.)
On the other hand $SPX$ is a product of Eilenberg - Maclane spaces. (corollary 4.84 there.)

Putting these two statements together you get what I wrote.