# Applications of the Dold-Thom theorem

Say $X$ has the homotopy type of a CW-complex. The Dold-Thom theorem states that $\pi_i SP(X) \cong \tilde{H}_i(X;\mathbb{Z})$, where $SP(X)$ denotes the infinite symmetric product of $X$.

I am just curious about some useful applications of this theorem or instances where this theorem simplifies calculations significantly.

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For one you can get the Mayer-Vietoris sequence in homology almost immediately from a homotopy pushout, but depending on what angle you look at it from, that may not be that useful (as in, to prove the theorem, you need to define the functor $H_*$ anyway, and then you probably already know it satisfies M-V).