# 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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## 1 Answer

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).

You can deduce a group structure on the Eilenberg-Mac Lane spaces, see this previous M.SE answer. (In fact it is probably possible to prove the existence of the Eilenberg-Mac Lane space using Dold-Thom)

At a philosophical level I can do no more than point you to Thomas Barnet-Lamb's excellent piece on Dold-Thom, and some of the reasons why it is important.

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Thank your for the reference to Thomas Barnet-Lamb's paper. I will take a look at it in more depth tonight. I know about the direct applications/corollaries (For example those in Hatcher) of the theorem but I am more curious about its usefulness in computations or other places where it arises. –  confusedmath Mar 23 '12 at 21:40